tag:blogger.com,1999:blog-37704195293891050612024-02-03T01:58:17.913-08:00Matholia - The World-class Mathematics Online PortalBlog for the Matholia mathematics learning portal - www.matholia.comMatholia Mathematicshttp://www.blogger.com/profile/12907740086274170195noreply@blogger.comBlogger53125tag:blogger.com,1999:blog-3770419529389105061.post-8987255327932818002023-05-08T21:42:00.005-07:002023-05-08T22:25:02.098-07:00Understanding Number Patterns and How to Avoid Mistakes When Identifying Them<p><b> Understanding Number Patterns and How to Avoid Mistakes When Identifying Them</b></p><div><div>A number pattern is a sequence of numbers that follow a specific rule or pattern. For example, the sequence of numbers 1, 3, 5, 7, 9 follows a pattern where each number is two more than the previous number. Identifying number patterns is an important skill in mathematics and is often used in fields such as computer science, finance, and engineering.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://matholia.com/" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img alt="Math_Number_Patterns" border="0" data-original-height="814" data-original-width="1500" height="217" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgMXi381UnddRqvohtqf0zXIjkNjexkMK_qkM5lo_KtkPma3x_UAOMfTcG7PkwUk4CX-xgUqYxE-E7fL49drVPScHmPCNk1FLykR85Q22qXqBhYlPdg_ttkc167sSWVbYxyUUKEQODWzER9lEbskdnyxr4qG8ylFidYUWao-e_acdgkMgnmmhXu9GVDQ/w400-h217/matholia_number_patterns.jpg" title="Understanding Number Patterns Matholia" width="400" /></a></div><br /><div><br /></div><div>When giving numbers in a pattern, it is important to be accurate and avoid making mistakes. Here are some tips to avoid giving the wrong number in a pattern:</div><div><br /></div><div>1. Check your work: After you have identified the pattern, double-check your work to make sure you have calculated each number correctly.</div><div><br /></div><div>2. Use a calculator: If the pattern involves complex calculations or large numbers, use a calculator to avoid making mistakes.</div><div><br /></div><div>3. Write out the pattern: Write out the pattern in full to ensure you haven't missed any numbers or made any mistakes.</div><div><br /></div><div>4. Look for errors: Once you have given the pattern, check for any errors by reviewing the sequence and comparing it to the pattern.</div><div><br /></div><div>5. Practice: The more you practice identifying and giving number patterns, the better you will become at avoiding mistakes.</div></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="335" src="https://www.youtube.com/embed/mZAjcrjVoas" width="403" youtube-src-id="mZAjcrjVoas"></iframe></div><br /><div><br /></div>HongYokMJ 5011379http://www.blogger.com/profile/06163971896340269736noreply@blogger.com1tag:blogger.com,1999:blog-3770419529389105061.post-92163824498034865072023-03-21T23:17:00.005-07:002023-03-21T23:19:40.857-07:00Busting the Big Number Myth: How Online Math Learning Can Help Young Learners Understand Number Value<p>One of the most common misconceptions that young learners encounter in numeracy is the belief that bigger numbers are always greater than smaller numbers. This misconception can arise because young learners often associate larger physical objects with "more" or "greater" than smaller objects. However, when it comes to numbers, this is not always the case. For example, 5 is greater than 3, but 0.3 is actually smaller than 0.5. This misconception can lead to difficulty understanding concepts like place value, fractions, and decimals. To help address this misconception, it is important to teach young learners about number relationships, the value of digits in different places, and how to compare numbers using symbols like greater than and less than.</p><p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinC4fnL5q97SC_82wVkOdAOuGZsg6vKfnsCJpHlbNkSWjC9yBXNu7DtfzURnSdTXuTVOCkd1WnxqYGZ8XlObQAh2_-3CjYLlNSCX0apbwAKYKojCu5OpT4m1_sPa23CjKm6sLPnmAtDLmCbR57bvq7sx5K180dPDJZ6fXqXbT9SzhvW2sVT0Ir8Tw6Ew/s5606/AdobeStock_365594859.jpeg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3737" data-original-width="5606" height="267" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinC4fnL5q97SC_82wVkOdAOuGZsg6vKfnsCJpHlbNkSWjC9yBXNu7DtfzURnSdTXuTVOCkd1WnxqYGZ8XlObQAh2_-3CjYLlNSCX0apbwAKYKojCu5OpT4m1_sPa23CjKm6sLPnmAtDLmCbR57bvq7sx5K180dPDJZ6fXqXbT9SzhvW2sVT0Ir8Tw6Ew/w401-h267/AdobeStock_365594859.jpeg" width="401" /></a></div><br /><p></p><p>An online learning math website with lots of instructional videos and dynamic practice, like <a href="https://matholia.com/">Matholia</a>, can help remedy the misconception that bigger numbers are always greater than smaller numbers in several ways.</p><p>Firstly, instructional videos can provide clear explanations of mathematical concepts that can help to dispel common misconceptions. The videos can explain the relationship between the size of a number and its value, and how this relates to concepts like place value, decimals, and fractions. By presenting information in a clear and accessible way, videos can help young learners to better understand the concepts they are learning.</p><p>Secondly, dynamic practice activities can provide opportunities for learners to apply what they have learned in a variety of contexts. For example, learners can practice comparing numbers in different forms, such as written form, numerical form, and visual form. By providing practice opportunities that are varied and engaging, learners can better internalize the concepts they are learning and build their confidence in their abilities.</p><p>Thirdly, interactive quizzes and assessments can help learners to identify and address their misconceptions. By providing immediate feedback and guidance, quizzes and assessments can help learners to correct their mistakes and build a deeper understanding of mathematical concepts.</p><p>Finally, online learning math websites can provide opportunities for learners to work at their own pace and in their own time. This can be especially helpful for learners who may struggle with math or who have different learning styles. By providing a flexible and supportive learning environment, online learning math websites can help learners to build their confidence and their skills.</p><p><b>An online learning math website with lots of instructional videos and dynamic practice can help remedy the misconception that bigger numbers are always greater than smaller numbers by providing clear explanations, varied practice opportunities, immediate feedback and guidance, and a flexible and supportive learning environment.</b></p>HongYokMJ 5011379http://www.blogger.com/profile/06163971896340269736noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-53145413552633633232023-02-23T17:48:00.001-08:002023-02-23T17:48:30.205-08:00The Bar Model: A Visual Problem-Solving Strategy in Singapore Math that Helps Learners Understand Math Better<p>The bar model is a visual problem-solving strategy used in <a href="https://matholia.com/">Singapore Math</a> that has gained widespread popularity among educators worldwide. It is a powerful tool that helps learners develop a strong conceptual understanding of math concepts by representing problems visually. By using bars or rectangular models to illustrate math problems, learners can break down complex problems into simpler, more manageable parts and make connections between different math topics.</p><p><b>How the Bar Model Works</b></p><p>The bar model is a problem-solving strategy that uses visual models to represent quantities and relationships between them. In a typical bar model, a rectangular bar is used to represent a known or unknown quantity. The bar is then divided into smaller sections, with each section representing a specific part of the quantity. For example, if the bar represents the total number of apples in a basket, then each section could represent the number of apples picked by each person.</p><p>The bar model is a versatile tool that can be used to solve a wide range of math problems, from simple addition and subtraction problems to more complex multiplication and division problems. By visualizing the problem, learners can better understand the relationships between different quantities and develop a deeper understanding of the math concepts being taught.</p><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://matholia.com/" imageanchor="1" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" data-original-height="643" data-original-width="902" height="310" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyGzTqcVl1jaqvT_izaaZD11MSG-1Wi6NKj0o_bH_4XMzcNAxg4mQb-JEyofIbdUJXdANv-K1rInWJIhvlI_ovk8nOaNiTOBgahbEJtwiAzXOUjXQYniAHpf5eWEX1uA08xneKZIB8kNZrGHGaMts7mmv5JlA9cm4Olfa4Fc0lZrkEq-ZdGdcVY5fwdw/w435-h310/Matholia_Addition_Bar_Model.png" width="435" /></a></div><br /><p><br /></p><p><b>How the Bar Model Helps Learners Understand Math Better</b></p><p>The bar model is an effective tool for helping learners understand math concepts because it enables them to:</p><p><b>Visualize the problem</b>: By representing math problems visually, learners can more easily understand the problem and the relationships between the quantities involved. This can help learners see the problem from different angles and develop a deeper understanding of the math concepts being taught.</p><p><b>Make connections between different math topics</b>: The bar model is a versatile tool that can be used to solve a wide range of math problems. By using the bar model to solve different types of problems, learners can make connections between different math topics and develop a more holistic understanding of math.</p><p><b>Develop critical thinking skills</b>: The bar model encourages learners to break down complex problems into simpler, more manageable parts. This helps learners develop critical thinking skills and improves their ability to solve problems independently.</p><p><b>Build confidence in math</b>: By using the bar model to solve math problems, learners can develop a deeper understanding of math concepts and build confidence in their ability to solve problems.</p><p>The bar model is a powerful visual problem-solving strategy that can help learners develop a strong conceptual understanding of math concepts. By representing math problems visually, learners can break down complex problems into simpler, more manageable parts and develop a deeper understanding of the relationships between different quantities. The bar model is a versatile tool that can be used to solve a wide range of math problems, and it encourages learners to develop critical thinking skills and build confidence in math. By incorporating the bar model into math instruction, educators can help learners understand math better and become more effective problem solvers.</p>HongYokMJ 5011379http://www.blogger.com/profile/06163971896340269736noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-52765272416026973172023-02-15T17:18:00.000-08:002023-02-15T17:18:09.623-08:00The Benefits of Adopting Singapore Math for Homeschooling<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXrT_ri2RgW5J8EX0f3dx7giAL4IooeWygIR3udpsKBmxefJp3BW_RR873tOgjWl7t3YWLeeGSLVDEz9VeqYkiuqcM-TqAQeIOcLK55gwtI70D8vhCMpH3QHn_aJL-rOgwMmJpS5ZcUnR-LNBPjPLc2HZ9v9BRO-cxtKVrr7t5lQX4yK3sl2DVfKC_-w/s5617/AdobeStock_524166211.jpeg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3746" data-original-width="5617" height="213" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXrT_ri2RgW5J8EX0f3dx7giAL4IooeWygIR3udpsKBmxefJp3BW_RR873tOgjWl7t3YWLeeGSLVDEz9VeqYkiuqcM-TqAQeIOcLK55gwtI70D8vhCMpH3QHn_aJL-rOgwMmJpS5ZcUnR-LNBPjPLc2HZ9v9BRO-cxtKVrr7t5lQX4yK3sl2DVfKC_-w/s320/AdobeStock_524166211.jpeg" width="320" /></a></div><p>For many homeschooling parents, teaching math can be a daunting task. Fortunately, there is a well-regarded math curriculum that has gained popularity around the world: Singapore Math. This curriculum is known for its emphasis on problem-solving, visualization, and building a strong foundation in mathematics. In this article, we'll explore some of the benefits of adopting Singapore Math for homeschooling.</p><p>Focus on problem-solving: One of the key strengths of Singapore Math is its focus on problem-solving. Students are taught to think critically and approach math problems from multiple angles. This emphasis on problem-solving not only helps students become more proficient in math, but also equips them with valuable skills that they can apply to other subjects and real-world situations.</p><p>Strong foundation in mathematics: Singapore Math is known for its strong foundation in mathematics. The curriculum is carefully structured to build a solid understanding of mathematical concepts, starting with the basics and gradually increasing in complexity. This approach helps students develop a deep understanding of math, rather than just memorizing formulas and procedures.</p><p>Emphasis on visualization: Another unique feature of Singapore Math is its emphasis on visualization. Students are encouraged to use visual aids, such as bar models and diagrams, to help them understand mathematical concepts. This approach helps students develop a more intuitive understanding of math, making it easier for them to apply what they've learned to new situations.</p><p>Clear and concise explanations: Singapore Math textbooks are known for their clear and concise explanations. The curriculum is designed to be easy for students to understand, with concepts introduced in a logical order and explained in a step-by-step manner. This makes it easier for parents to teach math to their children, even if they don't have a strong math background themselves.</p><p>Widely recognized and respected: Singapore Math is widely recognized and respected around the world. Many top-performing countries in math, such as Singapore and South Korea, use this curriculum in their schools. This recognition and respect can be a valuable asset for homeschooling parents, who can be confident that their children are receiving a high-quality math education.</p><p>Abundance of resources: Finally, there are many resources available for homeschooling parents who wish to adopt Singapore Math. There are numerous textbooks, workbooks, and other materials available for purchase, as well as online resources. <a href="https://matholia.com" target="_blank">Matholia</a> is an online primary mathematics portal based on the latest syllabus from the Ministry of Education, Singapore.The learning portal brings together a wealth of interactive content to ensure that pupils excel in all areas of primary mathematics whilst allowing teachers and parents the opportunity to closely monitor pupils' performance across the syllabus.</p><p>In conclusion, Singapore Math is a highly effective math curriculum that is well-suited for homeschooling. Its focus on problem-solving, strong foundation in mathematics, emphasis on visualization, clear and concise explanations, recognition and respect, and abundance of resources make it an ideal choice for homeschooling parents who want to give their children a solid math education.</p>HongYokMJ 5011379http://www.blogger.com/profile/06163971896340269736noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-33334208283180862282020-04-19T04:10:00.001-07:002020-04-19T04:10:15.281-07:00Counting on a Number Line (Missing Numbers)<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/GypeV6kUV3g" width="480"></iframe>Matholia Mathematicshttp://www.blogger.com/profile/12907740086274170195noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-14961468982829627002019-03-05T22:27:00.002-08:002019-03-05T22:27:24.518-08:00Problem Solving Task - Baking Cookies <span style="caret-color: rgb(32, 32, 32); color: #202020; font-family: Helvetica; text-align: justify;">For a school fundraiser, Nicole baked twice as many cookies as Rodney. Joey baked 5 times as many cookies as Rodney. If Joey baked 200 cookies, how many cookies did they bake altogether?</span><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOMJxqNePSO4Q1cNUQfwcJAoFBHgefbb67KHhTkxuMSLbA-McT_2vpF1Yp-h14-GEg7QxaXvD5FR_Ehw_56pb6oZ0iSBXKSLcIorA5HlLnOrHTcu1Ummz9ifpGD1XLZ1qJyzKD1MRto0FB/s1600/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="920" data-original-width="518" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOMJxqNePSO4Q1cNUQfwcJAoFBHgefbb67KHhTkxuMSLbA-McT_2vpF1Yp-h14-GEg7QxaXvD5FR_Ehw_56pb6oZ0iSBXKSLcIorA5HlLnOrHTcu1Ummz9ifpGD1XLZ1qJyzKD1MRto0FB/s640/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" width="360" /></a></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">Notes on the problem:</span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">This is a multi-step word problem that works well with a comparison bar model. </span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">When drawing the bar model, it’s helpful to show learners, “One time as many, two times as many…” while verbalizing, so that they can conceptualize multiplicative relationships rather than think of it as, for example five more bars for Joey. This avoids a common mistake of drawing the wrong number of bars in the model. </span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">Here is a sample bar model. In this one, the smallest unit (or baker with the least number of cookies) is identified first, and the relationships to that one are scaled up from there.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCm5vJCxgCFJOENuxSrgWNfo-cib_0Sxz6jh0B8LMx_-t2tqY-PRbiN_dM8gLsSXgcpejd4a6TmI0LY6t0tZKoW_JGlr3BQKYv5PtYGUJYEB3K67nFWLH4L5CFYepWQs13DkbxETcuK1U7/s1600/solution_1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="467" data-original-width="1592" height="186" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCm5vJCxgCFJOENuxSrgWNfo-cib_0Sxz6jh0B8LMx_-t2tqY-PRbiN_dM8gLsSXgcpejd4a6TmI0LY6t0tZKoW_JGlr3BQKYv5PtYGUJYEB3K67nFWLH4L5CFYepWQs13DkbxETcuK1U7/s640/solution_1.jpg" width="640" /></a></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">Once the model is successfully drawn, we can see that it can be solved in two or more steps.</span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);"><br class="Apple-interchange-newline" />One way to do it is to find the unit value by dividing what we know, the number of Joey’s cookies by the number of units he has: 200 ÷ 5 = 40</span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">Once that is found, the unit value can be populated into the model.</span></span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgilDNWnfSsgAD0j7mTpvkfb2kWTQ-m9iGlMmVeMNOQnkDDn90q9giiSmR4MXjsp0EghxY7Ga1uBo9EjexME0dGX3Erume0veu2tuqVSAaomH7tlbzPSkmqd5Ud4hkiweiOFEjm_UNDn6kz/s1600/solution_2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="470" data-original-width="1600" height="188" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgilDNWnfSsgAD0j7mTpvkfb2kWTQ-m9iGlMmVeMNOQnkDDn90q9giiSmR4MXjsp0EghxY7Ga1uBo9EjexME0dGX3Erume0veu2tuqVSAaomH7tlbzPSkmqd5Ud4hkiweiOFEjm_UNDn6kz/s640/solution_2.jpg" width="640" /></a></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">Concrete learners might write it into every unit, as shown, and then add up all the 40s or solve line by line and add 40 + 80 + 200 = 320.</span></span></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);">More abstract thinkers, who understand that 40 is the value of one part, and the whole consists of 8 equal parts, might jump straight to 8 x 40 = 320.</span></span></div>
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<span style="caret-color: rgb(32, 32, 32); color: #202020; font-family: Helvetica;">The second solution is shown below.</span></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9OGIolR0fyZ97jVDvnUxnSaH6P__uRLJuJnF-FVQ9tnoygFpfL6X634D8dW2cioQwdW8iEbF7jqyPSrr1p9PKpuB8Y6JV9XjBMFNsGbAw-m4SniYS1q6sNZLcI0nq2B0kwBIF3wp4ZPh-/s1600/solution_3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="461" data-original-width="1600" height="184" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9OGIolR0fyZ97jVDvnUxnSaH6P__uRLJuJnF-FVQ9tnoygFpfL6X634D8dW2cioQwdW8iEbF7jqyPSrr1p9PKpuB8Y6JV9XjBMFNsGbAw-m4SniYS1q6sNZLcI0nq2B0kwBIF3wp4ZPh-/s640/solution_3.jpg" width="640" /></a></div>
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<span style="color: #202020; font-family: Helvetica;"><span style="caret-color: rgb(32, 32, 32);"><br />The baked 320 cookies altogether.</span></span></div>
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
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</style>Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-27161792659534557392019-02-26T17:39:00.000-08:002019-02-26T17:39:23.441-08:00The Importance of Problem Solving in MathWhen we look at math accomplishment around the world, some countries stand above others in their achievements. Singapore’s record is outstanding, for example, often taking the number one spot in international testing in reading, science and, of course, math.<br />
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Digging deeper into the reasons for Singapore’s success, a pattern emerges. This young country has invested most of its resources in developing a high-quality approach to education. Part of this includes drawing from the best practices from around the world in all subjects. Professionals are sent around the world to study the research and practices in different countries and bring their findings back to the ecosystem of their teacher training.<br />
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Another reason for their success is the emphasis on problem solving. Where historically math has been taught at elementary levels as primarily computation practice, Singapore was a leader in changing the focus to problem solving. Not only that, but they took from the work of Jerome Bruner’s <a href="https://www.amazon.com/Process-Education-Bruner-1-Jul-1960-Paperback/dp/B012HUZOGU/ref=as_li_ss_tl?ie=UTF8&qid=1543434576&sr=8-3&keywords=bruner+1960&linkCode=sl1&tag=aspiarts-20&linkId=02e8015ce95e5638224dc15ba5c6a435&language=en_US">The Process of Education</a> (1960) to develop their approach of introducing concepts first through concrete experiences, then pictorially, and then finally the abstract or procedural level. This enables a robust form of learning that supports long-term learning, not only memorization or working memory.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpontZ64okvIzMzYZ1orPpsBBCVrY-JLpWHMETB_y9wjChgpbd6Aj4Yz70GJARQTU8b2JZCBIWkzUrPp2TQqR5UiW9pct1SB-R6jYOboU6ncJ6RcKcU7Q6LZHCGMz1bCCYbEafNw_J3-Ay/s1600/0c9c277b-3046-4303-bfe7-04fa5cbbdbea.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="467" data-original-width="847" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpontZ64okvIzMzYZ1orPpsBBCVrY-JLpWHMETB_y9wjChgpbd6Aj4Yz70GJARQTU8b2JZCBIWkzUrPp2TQqR5UiW9pct1SB-R6jYOboU6ncJ6RcKcU7Q6LZHCGMz1bCCYbEafNw_J3-Ay/s400/0c9c277b-3046-4303-bfe7-04fa5cbbdbea.jpg" width="400" /></a></div>
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Then in the 1990s, a teacher in Singapore created an ingenious approach to problem solving called bar modeling, or tape diagrams in US Common Core parlance. This approach allows a systematic way to solve word problems that is far more efficient that previous strategies, which would often include the tedious guess-and-check approach.</div>
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Bar modeling also supports algebraic thinking, in which the unknown is identified, usually with a question mark. That model then allows powerful visualization in problem solving, so when students make a bridge to algebra, it is demystified and scaffolded.</div>
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One reason students worldwide might not succeed in math is because they are not challenged in the <i>zone of proximal development</i> (Vygotsky). They are either given tasks that are too challenging, creating frustration, or too easy, leading to boredom. Also, when teachers are unable to challenge students at the appropriate levels, the temptation is to give students too much help too quickly, or give them the answers. This undermines students’ ability to struggle and persevere in problem solving.</div>
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This is connected to the mindset work of Carol Dweck and its application to math by Jo Boaler and her team. When struggle is seen as negative and the right answer is paramount, students feel like failures when they struggle. When productive struggle is encouraged, though, and the process, rather that the answer, is emphasized, new patterns emerge. Students persevere, feel successful when they work hard, develop more confidence, and are more willing to take risks in learning. They embrace mistakes as learning opportunities and become more supportive of their classmates. These abilities then carry on through their school careers and life. This is why the first Standard for Mathematical Practice adopted in the US is “Make sense of problems and persevere in solving them.”</div>
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Therefore, it’s extremely important to include challenging word problems, and the appropriate scaffolds for student success, as part of students’ math experiences.</div>
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The Matholia bar modeling tools and practice activities can fulfill these needs, and the math problems released in the Matholia Loop newsletter have their place in the classroom as well.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvGYVUJvj0fGhKwkMrBf39v1GgX5Wo6KioXfYWF6Reh4EvDl7A3nMTjbkoSNe8afX_ETFR6z78kO5RuBynL4rtMGKr-PIaQCnmcy8tHcU7XzqG67uNxun3qf4GaxhqTlp6tLtkJ-51X851/s1600/e0e2401d-aaa7-4544-9c97-3bd93b65bd37.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="601" data-original-width="800" height="300" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvGYVUJvj0fGhKwkMrBf39v1GgX5Wo6KioXfYWF6Reh4EvDl7A3nMTjbkoSNe8afX_ETFR6z78kO5RuBynL4rtMGKr-PIaQCnmcy8tHcU7XzqG67uNxun3qf4GaxhqTlp6tLtkJ-51X851/s400/e0e2401d-aaa7-4544-9c97-3bd93b65bd37.jpg" width="400" /></a></div>
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Taken from the Practice environment at <a href="http://matholia.com/">matholia.com</a></div>
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-11200758705394343602019-02-26T17:25:00.000-08:002019-02-26T17:25:17.194-08:00Problem solving TaskFind the value of the shapes.<br />
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Solve for each shape value. This puzzle is appropriate for grade 3 and up.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLVb-VuJuUHtwAvHTHVPuo6cB3Jpb2dVNBgG4y0FQ1ZBoxmYTdz72tIUQv8ZfBJ8X_iywoAbxsecc6rgzsHDpyM6G279TQZSufE-t21x1m3It5HGkvPSjP5S5BskThI2y181DqG0QInTw7/s1600/d60fedf9-12a3-49aa-8f3e-c5e92d532206.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="532" data-original-width="645" height="262" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLVb-VuJuUHtwAvHTHVPuo6cB3Jpb2dVNBgG4y0FQ1ZBoxmYTdz72tIUQv8ZfBJ8X_iywoAbxsecc6rgzsHDpyM6G279TQZSufE-t21x1m3It5HGkvPSjP5S5BskThI2y181DqG0QInTw7/s320/d60fedf9-12a3-49aa-8f3e-c5e92d532206.jpg" width="320" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s1600/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="920" data-original-width="518" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s640/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" width="360" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLqT0_UfBfLkD08C50vv4gf5e05A9hRHGbofXkPdAs9KVtk4RKwVxvMQcOCJMTjhw6TATaYmS_rp0kY3vSXGHSzSnB5wNKb20D1aA7nxxVPRByrtacUyKf65Pus-Gm4-AgZ8MX1zcphKSQ/s1600/Screen+Shot+2019-02-27+at+11.23.17+am.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="83" data-original-width="481" height="55" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgLqT0_UfBfLkD08C50vv4gf5e05A9hRHGbofXkPdAs9KVtk4RKwVxvMQcOCJMTjhw6TATaYmS_rp0kY3vSXGHSzSnB5wNKb20D1aA7nxxVPRByrtacUyKf65Pus-Gm4-AgZ8MX1zcphKSQ/s320/Screen+Shot+2019-02-27+at+11.23.17+am.png" width="320" /></a><br /></div>
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Notes on the problem:</div>
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This puzzle reflects the algebraic thinking supported and encouraged throughout the Singapore Math curriculum. By recognizing that three of the same shape require a total quantity to be distributed equally between them, students make a connection to the concept of partitive division. They then transfer the value of one unit to find the value of the others. This allows the development of flexible problem solving in multi-step problems. It is related to the thinking used in bar model strategies for word problems.</div>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-34168684335581152602019-02-26T17:17:00.000-08:002019-02-26T17:17:05.127-08:00The Importance of the Unit, Part II: Fractions as Units<div style="text-align: justify;">
How many times have we heard, “I hate fractions!” from both children and adults? So many times, because after students spend years learning how numbers work, suddenly they are introduced to a new kind of number that breaks all the rules they learned before.</div>
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When we add or subtract fractions, we don’t add them the way we add whole numbers, but we have to remember a whole set of new rules. And when we multiply by a fraction, sometimes the product is bigger, and sometimes it’s smaller.</div>
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Dividing fractions makes no sense -- so students learn “keep-change-flip,” which if you ask students (or teachers) to explain why it works, you will get blank stares and stuttering. Not only that, but many students don’t remember which fraction to “flip,” or why to flip the divisor and not the dividend. But this mnemonic is so popular that people create videos and other funny ways to remember it and how to apply it. Still, errors abound.</div>
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And then there are “improper” fractions too -- and if you ask some students, that term should apply to all fractions, because they don’t follow proper number rules!</div>
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So as soon as decimals appear on the horizon, many students will avoid using fractions at all costs, and will run straight for the calculator and converting to decimals in high school. This makes calculating slope, ratios, and a number of other concepts more challenging, concepts that would be easier with a strong grasp of fractions.</div>
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THERE IS A BETTER WAY!</div>
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Unlike students who learned in procedural ways, in which operations with fractions appear mysterious and illogical, when students learn them conceptually, fractions often become their favourite numbers in math. How does this work?</div>
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In early grades, students are introduced to the basic fractions they are most likely to encounter in everyday life: halves, thirds, and fourths. These are introduced in Singapore Math using manipulatives like fraction tiles and circles, and are then mirrored with pictures of these units. Students are given stories involving fractions that relate to real life as well, such as sharing pizza or other equally divisible items. The Fraction Tiles and Fraction Discs tools on Matholia can be used to model these units.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS2sD0uq7OG8XxvnkKzQ3cOpkRgzbr6Ln_rmWzycWQw9s42_C23opqFeoK3Ot_bQF017c2xv618nQFMZCcFOT4QWg0n-DJAE-Xa9SxK9lNtGmmoiVygkXP56KB47CrDWreTIc3qd8QU6hC/s1600/d4c2d903-db95-46b1-a531-0a63231218b4.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="539" data-original-width="719" height="297" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiS2sD0uq7OG8XxvnkKzQ3cOpkRgzbr6Ln_rmWzycWQw9s42_C23opqFeoK3Ot_bQF017c2xv618nQFMZCcFOT4QWg0n-DJAE-Xa9SxK9lNtGmmoiVygkXP56KB47CrDWreTIc3qd8QU6hC/s400/d4c2d903-db95-46b1-a531-0a63231218b4.gif" width="400" /></a></div>
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When students are introduced visually to fractions, they are then given the concept that a fraction is a unit just like any other unit. They have been introduced to counting by ones, tens, apples, and so on; counting by halves or thirds is just counting by another type of unit.</div>
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Using a number line, students can then count by units in later grades to find non-unit fractions and mixed numbers. For example: one-fourth, two-fourths, three-fourths, four-fourths (or one whole), five-fourths , etc.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim2r1pbENBwxwA2JlVs59-5-LPlLV7v6ymGF60uanhAwe5uoQV_P3Nt5OltOWvyIDND0MY7abiZQlqpgDNzo3hGfTwprjh5WbkxoYNAm0FfN5KVrjNDSgcscK1O7CS8tdSfDEUtOHZpiEi/s1600/6d21bf48-21da-4b4f-a2ff-c86e85bfd363.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="539" data-original-width="719" height="298" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim2r1pbENBwxwA2JlVs59-5-LPlLV7v6ymGF60uanhAwe5uoQV_P3Nt5OltOWvyIDND0MY7abiZQlqpgDNzo3hGfTwprjh5WbkxoYNAm0FfN5KVrjNDSgcscK1O7CS8tdSfDEUtOHZpiEi/s400/6d21bf48-21da-4b4f-a2ff-c86e85bfd363.gif" width="400" /></a></div>
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Students practice converting back and forth between different forms, including equivalent fractions and mixed numbers. This makes adding and subtracting more intuitive, because the big idea is -- you have to have the same unit to add or subtract! In other words, how to you add two apples and three oranges? You can’t -- but you can convert both to fruit, and then you can add.</div>
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In the same vein, how do you add ⅓ and ⅖? You don’t -- but you find the same value in an equivalent unit, in which you can then add or subtract.</div>
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In future articles, visual ways to find equivalent fractions, and the way this approach relates to multiplication and division of fractions, will be explored.</div>
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-82232564104582623712019-02-12T16:30:00.000-08:002019-02-12T16:31:20.657-08:00Building a Craft Project<div style="text-align: justify;">
Jae needs to cut wood for a craft project. The instructions provide the size relationships, and Jae gets to decide the actual size. Her first piece of wood will be 𝒙 centimeters long. The middle part has to be 3 times as long as the first. The final part is 11 cm longer than the middle part.</div>
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a) Express the length of the final part in terms of 𝒙.</div>
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b) If the first piece of wood is 25 cm long, how long will the final part be?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s1600/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="920" data-original-width="518" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s640/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" width="360" /></a></div>
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A problem like this would likely be found in Primary 6 or US sixth grade math. Students are encouraged to make the connection between bar modeling and variables.</div>
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A unit in a bar model is pretty much just like a variable; it is another way to represent an unknown. Students who have experienced bar models and part-whole thinking are more likely to succeed when it comes to solving for variables.</div>
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If students struggle with representing the situations, they can start by bar modeling the problem. One sample solution could be like this. After the model is created, just place the 𝒙 in the appropriate bars.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-ywo9MN9maoqPQq0qXQQGnRGFB6iQBm0ypCBRO57brkSwPBXneAiPKbCVyLGKMjJxk2BMO3iJ0qpRbrO0w-ZOmfaq9pl-fpSLRJjiTbdyHCVaANbUwJlnO1T5d-Ft5uW48Wo-yaRW0fks/s1600/barmodel.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="496" data-original-width="1239" height="256" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-ywo9MN9maoqPQq0qXQQGnRGFB6iQBm0ypCBRO57brkSwPBXneAiPKbCVyLGKMjJxk2BMO3iJ0qpRbrO0w-ZOmfaq9pl-fpSLRJjiTbdyHCVaANbUwJlnO1T5d-Ft5uW48Wo-yaRW0fks/s640/barmodel.jpg" width="640" /></a></div>
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For students who need support making a connection, ask questions like these:</div>
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<li style="text-align: justify;">How many times do you see 𝒙?</li>
<li style="text-align: justify;">How do we write it when we have 3 times something?</li>
<li style="text-align: justify;">Imagine is a pencil. How could you write it with pencils instead of 𝒙?</li>
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A student who finds abstract thinking easier to access might come up with the following solutions immediately:</div>
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a) 3𝒙 + 11, or 3•𝒙+ 11, or 3(𝒙) + 11</div>
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b) 3(25) + 11 = 86</div>
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The final part is 86 cm long.</div>
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Did you or a student of your solve it a different way? Share the strategy with us!</div>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-31302215085838197602019-02-06T20:26:00.001-08:002019-03-05T21:45:04.237-08:00Buzzwords Decoded: Number Talks and Mental Math<div style="text-align: justify;">
One challenging thing about teaching is that terminology and popular buzzwords change all the time. For example, “math talks” or “number talks” used to commonly refer to a conversation about math in the classroom. These days, though, “number talks” usually refer to a specific protocol for a structured conversation about mental math, as outlined in the book <a href="https://amzn.to/2FOBNsY">Making Number Talks Matter: Developing Mathematical Practices and Deepening Understanding, Grades 4-10, by Cathy Humphreys and Ruth Parker (2015). </a></div>
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Why has this protocol taken over this terminology? One reason is that the protocol is effective at reaching all students and achieving multiple goals in developing mathematical thinking. </div>
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The protocol begins with students being presented a mental math problem. Students place their fist over their heart. When they have solved it using one strategy, they open one finger. Two strategies, another. And so on. The teacher gives a set amount of time, and students find as many strategies as they can. </div>
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After one or a series of problems are solved, a class discussion of strategies ensues. Students share the strategies they used to find the solution. </div>
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Why is this effective?</div>
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<b>Wait time.</b> In many traditional math classrooms, teachers respond to raised hands, often among the first ones up. This creates several problems:</div>
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<li style="text-align: justify;">Boys have been shown to raise their hands even if they aren’t sure of the answer, or even if they don’t have an answer, while girls are <a href="https://www.nytimes.com/2017/10/31/opinion/im-10-and-i-want-girls-to-raise-their-hands.html">more reluctant to do so</a>. This can lead to an imbalance of learning and to girls feeling less confident in class.</li>
<li style="text-align: justify;">Some students process more slowly than others. If the fastest students are consistently providing the answers, those are the ones seen as “smart,” while other students may have more creative or innovative solutions. This can lead to slower/deeper students checking out mentally in class.</li>
<li style="text-align: justify;">When one student is called on at a time, this leads to one student learning at a time when it’s a simple problem. If it’s a question of sharing strategies or ideas, it works to have students share one at a time, because that’s interesting! But for numerical or simple solutions, other strategies, such as choral response, white boards, or Plickers, can be more effective to collect responses and retain student engagement.</li>
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<b>Differentiation.</b> The activity is naturally differentiated. </div>
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<li style="text-align: justify;">Students who know the answer by rote can find other ways to find the same solution.</li>
<li style="text-align: justify;">Students who are still learning the concept have more time to come up with one strategy.</li>
<li style="text-align: justify;">Everyone has a chance to feel successful. Even the teacher can participate and learn something.</li>
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<b>Rich learning possibilities.</b> When students share their strategies with each other, they may learn much more than with traditional approaches to teaching mental math.</div>
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<li style="text-align: justify;">Students who have made errors can correct themselves and understand why they made that error, possibly preventing the same error in the future.</li>
<li style="text-align: justify;">Students who did not understand a strategy when it was taught in class might understand it when another student presents it.</li>
<li style="text-align: justify;">Spending time on multiple strategies can help create stronger neural networks around the concept, making reconnection to the pattern easier in the future.</li>
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<a href="https://www.matholia.com/au/apps/tools/mt_tlsu4_0jm7w?cid=603">The Matholia tools</a> can be useful to present a situation for students to consider, or for students to represent their thinking. For example, students who have learned how to add fractions without common denominators might be asked to find the sum of these two fractions. </div>
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Alternatively, they could be presented with 3 ÷ 4 and asked to use these tools to represent this fraction as a sum of two or more fractions. </div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHgDxAIF2uF0RiP0StiRSq5Q_frJPhDgTunPNNMjyLz0aVEWf2AMfcUZf-8uGMSwFydR8ldzgrX7f4FupK8EAGIfkTswACgA_YDFvRkpYeT_rizzCHoxDU0sxUzEESZpnGsFmf0bhJ9U3z/s1600/article_10_fractions_tool.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="540" data-original-width="960" height="360" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHgDxAIF2uF0RiP0StiRSq5Q_frJPhDgTunPNNMjyLz0aVEWf2AMfcUZf-8uGMSwFydR8ldzgrX7f4FupK8EAGIfkTswACgA_YDFvRkpYeT_rizzCHoxDU0sxUzEESZpnGsFmf0bhJ9U3z/s640/article_10_fractions_tool.gif" width="640" /></a></div>
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One example of how a number talk might look in a classroom can be seen here. Note that the fist and fingers response is not shown, but the solution strategy sharing is.</div>
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Have you tried this approach? Share your experiences with us!<br />
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-20627785181928709992019-02-03T17:05:00.000-08:002019-02-03T17:05:03.335-08:00Teaching with Anchor Tasks <div style="text-align: justify;">
What are anchor tasks, and what place do they have in the classroom?</div>
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To understand this, we need to begin with a little neuroscience. According to Dr. David Sousa, the “primacy” effect of the brain means that in a lesson, the first ten minutes are the most important for learning. The brain is able to retain the most in this first period. This means that if teachers waste this with attendance, handing in homework, reviewing old work, or other administrative tasks, they are losing the most valuable time in their lesson!</div>
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How can we leverage this knowledge more effectively? Singapore teachers have done so by creating anchor tasks. These are rich tasks that engage learners in discovering concepts through exploration and application and are accessible to multiple levels at once. </div>
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A mathematical problem or situation is presented to students to explore and question, in which the students work in groups and the teacher facilitates. A rich discussion ensues, in which different learners share strategies and learn from each other, and the teacher can facilitate struggling learners to make sense of the concept.</div>
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One example is a third grade teacher preparing a class for a class on multiplication. The students are familiar with multiples of 2, 5, and 10. Today she wants to introduce multiples of 3, so she projects this image and gives a copy of this chart to each table.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo6VTqiq2kMC37yrK-xoWPoiAoIDM0IyeX3U4U31-p09CdDsiMQ_UTerbh61sQEeYp7T6cAFbgaP-E9T6U3Tbifwpugm3y2c8Z2Uakf5D7p_8PSxky1eekngk3bFnrJffywBKYJRQznS9m/s1600/9fac1060-ebcf-410f-b338-85d1606f8a9e.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="367" data-original-width="1600" height="146" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo6VTqiq2kMC37yrK-xoWPoiAoIDM0IyeX3U4U31-p09CdDsiMQ_UTerbh61sQEeYp7T6cAFbgaP-E9T6U3Tbifwpugm3y2c8Z2Uakf5D7p_8PSxky1eekngk3bFnrJffywBKYJRQznS9m/s640/9fac1060-ebcf-410f-b338-85d1606f8a9e.jpg" width="640" /></a></div>
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She also provides manipulatives, Rekenreks, and white boards to the students to support their explorations.</div>
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She then asks, “Can you fill in this the missing blanks on this chart? Do it as a team, and share your thinking about how you were able fill in the blanks.”</div>
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This will encourage pattern thinking and extension. She may also encourage early finishers to extend the chart and fill in as many blanks as they can, justifying their work with pictures or demonstrations to avoid relying on rote memorization.</div>
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One student might make doubles of each number to find the product on the second row, and then another might recognize the commutative property and how that leads to the related product. For example, the student might demonstrate that 3 twos are 6 by laying out 3 rows of 2. Another student might then point out that this is the same as 2 threes if you switch the rows and columns. This leads to an intuitive understanding of the commutative property.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnVeTLpD0vx5tk4epJftx2obmBCYtc_lsfKTfCSrE8qDm2yN5ShyphenhyphenvNsEzfSflvM9XZ5eLoF0A8GxopGYyZb3WrKdHo-5xvauPXRIenb7D165HFHSdFYxvR-A9-YwcbLKly8wUhCoehUBC4/s1600/article_10+2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="429" data-original-width="642" height="426" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnVeTLpD0vx5tk4epJftx2obmBCYtc_lsfKTfCSrE8qDm2yN5ShyphenhyphenvNsEzfSflvM9XZ5eLoF0A8GxopGYyZb3WrKdHo-5xvauPXRIenb7D165HFHSdFYxvR-A9-YwcbLKly8wUhCoehUBC4/s640/article_10+2.gif" width="640" /></a></div>
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When an anchor task is used to launch a lesson, students are fully engaged and learning, and all students are involved, rather than just a few vocal ones. The practice in the workbook or online then has meaning and relevance, rather than being dry and disconnected. </div>
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Do you have experience with anchor tasks in your classroom? Share it with us in the comments!</div>
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-29644872583545912032019-02-03T16:39:00.000-08:002019-02-03T16:39:14.696-08:00Problem Solving Task<div style="text-align: justify;">
Honora gave 7 candies each to 4 of her friends. She had 37 candies left. How many candies did Honora have at first?</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s1600/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="920" data-original-width="518" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvI5B3Su6zlc6od8XZgbi_Bv4w-ZQvpmuyUXifSvDw5B-3-x18BW-ybexaFoUOtSXcEXHxmBTRVgwVLdHGrBeQltHFsxEbJgzS3h2fhSl0m-8mNNNb51xTP4iueuYAXMPutSZMYk26RR27/s640/41ad6493-25ab-4e9f-8b01-73ee4c852505.jpg" width="360" /></a></div>
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This is a mixed-operation problem with multiple steps. There are several ways to approach this, but a common mistake might be to subtract from 37 and then divide or multiply. This is often seen when students focus too much on the numbers and not the meaning. This is one way bar models can be very helpful, as it causes a student to pause and think about the meaning of what is happening.</div>
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Since this is a part-whole question, a first model </div>
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might look something like this:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEida_e7333FnJrlHVznTn1Ow1LhF1TWQmUrcvTaHO0wg9zcMpBVU318AXQKC_3x3NdVWzuHARwTF08DefQz9k5XBHZ6OWAV1MsyWQbKGs-dJytImmEqPZopXK9ahTZjo0KcnI0PYZs2SMVi/s1600/002605df-30ed-4862-8848-5abd0770f611.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="455" data-original-width="1600" height="180" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEida_e7333FnJrlHVznTn1Ow1LhF1TWQmUrcvTaHO0wg9zcMpBVU318AXQKC_3x3NdVWzuHARwTF08DefQz9k5XBHZ6OWAV1MsyWQbKGs-dJytImmEqPZopXK9ahTZjo0KcnI0PYZs2SMVi/s640/002605df-30ed-4862-8848-5abd0770f611.jpg" width="640" /></a></div>
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Then the student might identify the candies given to friends, like so:<br /><br /><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTp21I0BHTpD27GaYTGJfz6y01pGFr88d5m8jZUQjncaJWOABCp6h1KDQRHqxVDuIS795j1svwEe4aRtKPmisRYjiBbst0kPGz56_kOjPpOrfQhfHwukyTGdAbO9RdF4D6hWf0UIPEncMP/s1600/1ada3823-b3b5-4b36-b327-2611305974af.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="455" data-original-width="1600" height="182" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTp21I0BHTpD27GaYTGJfz6y01pGFr88d5m8jZUQjncaJWOABCp6h1KDQRHqxVDuIS795j1svwEe4aRtKPmisRYjiBbst0kPGz56_kOjPpOrfQhfHwukyTGdAbO9RdF4D6hWf0UIPEncMP/s640/1ada3823-b3b5-4b36-b327-2611305974af.jpg" width="640" /></a></div>
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The student now has both parts, and all that is left is to add them to make the whole and identify the quantity for the question mark: </div>
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28 + 37 = 65, so</div>
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Honora had 65 candies at first.</div>
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Did you or a student of yours solve it a different way? Share the strategy with us!</div>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-18367767599996183472019-01-23T20:07:00.000-08:002019-01-23T20:07:07.315-08:00Mental Addition and Subtraction Strategies<div style="background-color: white; color: #202020; font-family: Helvetica; font-size: 16px; line-height: 24px; margin-bottom: 10px; margin-top: 10px; padding: 0px; text-align: justify; text-size-adjust: 100%;">
Have you ever met someone who could do amazing computations in their head, faster than you could even pull out a calculator?<br /><br />Conversely, have you seen older students or adults who still have to add and subtract by counting up or down from the starting number?<br /><br />One way Singapore Math differs from other curricula is that it takes what people who intuitively figure out how numbers work and can calculate mentally, and makes it explicit so every student can use these same strategies.<br /><br />The approaches begin in early grades, such as Kindergarten and first grade, where children go beyond knowing the counting sequence and learn the different ways any number within 10 can be decomposed. For example, they practice with:<br />5 is 4 and 1<br />5 is 1 and 4<br />5 is 3 and 2<br />5 is 2 and 3</div>
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These are first introduced with decomposing with objects, and then those are related to number bonds with objects, both in concrete and picture form, then to number bonds with pictures, and finally to equations, first in word form as above, then with symbols.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHRZ_7K_gQth5Q8xhmsmkiaGFa47uhJF7f2N0s9OuRbOlUR611gXCXZ596RWFEBrYexNHcGC3o5VDsh7Bc-lH-Dp8nbHgZDwIovzKEzL33D_MpcKyHGZiGN-hHoxQ1m4fHiY4Rbm7vlcxV/s1600/Screen+Shot+2019-01-24+at+1.09.48+pm.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="774" data-original-width="353" height="640" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHRZ_7K_gQth5Q8xhmsmkiaGFa47uhJF7f2N0s9OuRbOlUR611gXCXZ596RWFEBrYexNHcGC3o5VDsh7Bc-lH-Dp8nbHgZDwIovzKEzL33D_MpcKyHGZiGN-hHoxQ1m4fHiY4Rbm7vlcxV/s640/Screen+Shot+2019-01-24+at+1.09.48+pm.png" width="290" /></a></div>
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When number bonds are introduced, part-whole thinking can develop, leading to ease with both composing (adding) and decomposing (subtracting). Both the part-whole models and bar models build upon that.<br /><br />Fluency with these compositions within ten provides the foundation for all the subsequent computations. For example, if each subsequent decade is understood as 10 or a multiple of 10 plus one of the known compositions, it’s a simple matter to apply the same patterns. No longer is each number a discrete entity; each is made up of the same type of puzzle piece as the ones they have already learned.<br /><br />So students learn to apply the rules they learned to compose and decompose with larger place values, using different manipulatives to become acquainted with the values of the different digits and how they work together. This leads to more complex strategies starting in second grade, such as addition and subtraction compensation.<br /></div>
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Addition Compensation</h4>
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One form of addition compensation is the “make a ten” strategy. This is where one addend is decomposed to make a ten, or a multiple of ten, to make adding easier. Some examples of strategies are the number bond decomposition and the “arrow way,” or step-by-step approach. For example:</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2GqbbwELUW84vj4mCqcZqc3hFTG8G7zjlRSsi-C6CvcYn09tWdWKY6QprYFcpO-ZjcVRnSFBpYGKE5Df-TGizfmtTPc3VxpEF3s3jdm5V0cYVm8n7NrRrpOCOH68P6IGo4nsDRsx_iud8/s1600/kid1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1190" data-original-width="881" height="400" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2GqbbwELUW84vj4mCqcZqc3hFTG8G7zjlRSsi-C6CvcYn09tWdWKY6QprYFcpO-ZjcVRnSFBpYGKE5Df-TGizfmtTPc3VxpEF3s3jdm5V0cYVm8n7NrRrpOCOH68P6IGo4nsDRsx_iud8/s400/kid1.jpg" width="295" /></a></div>
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<strong id="docs-internal-guid-82bfe9ac-7fff-5ba3-5e62-8dd6f1452ae1">Subtraction Compensation</strong></h4>
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This depends on the understanding of subtraction as finding the missing part of a whole and as a difference. The strategies are a bit different here. Number bonds cannot be used in the same way; instead, the students learn that to find the difference, the same number can be added to both minuend and subtrahend, and the difference will stay the same. Two examples here are making a ten out of the subtrahend, and step-by-step subtraction.</div>
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There are so many other strategies and so many ways to use these strategies. Sharing solution strategies can be a great teaching tool. Both my students and I have gained flexibility and ease with numbers the more we practice with these. We can also mentally check our answers if we use an algorithm or a calculator, always checking, “Does that make sense?”<br /><br />Do you use a similar or alternate strategy? Try these out or share your own!</div>
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<span style="font-size: 12px;"><em>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at susanmidlarsky.com.</em></span></div>
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<span style="font-size: 14px;"><em>The Matholia Team</em></span></h1>
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-74223993813296605192018-12-04T17:32:00.002-08:002018-12-04T17:33:13.859-08:00Problem Solving Task<h2 style="text-align: center;">
Problem Solving Task</h2>
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Alijah is trying to mark the height of an aquarium. She also wants to mark different heights of the aquarium to ensure it holds enough water for her fish. She knows that when it is full, it holds 24 litres of water. She also knows that the base is 30 cm wide and 40 cm long.</div>
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(a) Find the height of Alijah’s aquarium.</div>
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(b) Using a piece of string, how can Alijah mark the heights when the aquarium is ¼, ½ and ¾ full?</div>
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Solution and Discussion:</h3>
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This is a fairly typical fifth or sixth grade volume decomposition problem. It involves several concepts and skills:</div>
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● Conversion from cm to cubic cm to litre, and back again. 1 L = 1 000 cubic cm</div>
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● Area x height = volume, and the relationship of straight, square, and cubic units</div>
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● Understanding of volume as a part-part-part-whole problem, that is, length x width x height = <span style="color: white;">ghjk</span>volume, and any decomposition of that</div>
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● Measurements and fractions of measurements</div>
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(b) Alijah can place a piece of string from the base to the top of the aquarium. She can then fold the string in half to find the ½ full mark. Repeating the process again she can mark the ¼ full mark. Finally, by placing the ¼ length string at the ½ way mark, she can find the ¾ full mark.</div>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-15697014627085831222018-11-21T19:56:00.000-08:002018-11-21T19:57:30.376-08:00Conceptual Approaches to Teaching Division<div style="text-align: justify;">
When observing a fifth grade teacher the other day, I noticed that while the students were engaged in many positive ways, they relied upon DMSB -- Daughter, Mother, Sister, Brother -- or some other mnemonic to remember the procedure for long division. They were also ignoring the value of the unit they were dividing. This is very common, but there are better ways, as Singapore Math shows us! </div>
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Why avoid mnemonics like these to help divide? </div>
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● They lead to place value errors, because all digits are treated the same. </div>
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● It’s no easier to remember these than an actual procedure. </div>
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● Students can remember them in the wrong order. </div>
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● They reinforce the idea that math is “magic,” following rules that no mortal can understand. </div>
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How can division be taught conceptually?</div>
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There are multiple approaches that can lead to the understanding of division. In early grades, students take a number of objects and split it into equal groups. A counter such as the unit cube in Base Ten blocks can be used, and the Matholia tool can help with this.<br />
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Work with arrays and place value disks can reinforce the idea of division as repeated subtraction as well. So can using bar models, as illustrated in the article <a href="http://matholia.blogspot.com/2018/10/bar-modelling-for-two-types-of-division.html">Bar Modelling For Two Types of Division.</a></div>
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Some math approaches undercut conceptual strategies by always representing the problem horizontally or in division’s case, using a division bracket. Starting by representing the problem horizontally allows more flexibility in thinking and does not guide the student into immediately jumping into an algorithm. The problem can always be rewritten in algorithm form if desired.</div>
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Once the students reach a solid understanding of what is happening in division, the division can be taught in multiple ways. What is commonly considered the division algorithm is just one of several, and not even the most efficient. Two algorithms that can be used to scaffold the understanding of the common long division algorithm are:</div>
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<i>Partial Quotient Division -- layout 1:</i></div>
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In this style, students use whatever math facts they know to divide the numbers. They then continue to subtract and perhaps repeat the same fact multiple times in order to reach their goal. Then the partial quotients are added to find the total quotient.</div>
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<i>Partial Quotient Division -- layout 2:</i></div>
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This style resembles traditional long division more closely. Eventually they will be able to combine steps to find the total quotient.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKBpfKdztdgpLCpxQUa14dzGvHbUBB5r_VBi7PkXUCk0ZwkuvMqlsJV7dffa_Fj8woM3fq5ciz-GPOBgPdXtvuNp0Va2LAI7w4KsTBDK_FHhfl0clpaM8hcJYJBXE8H5GaHxNY9HC1afSW/s1600/Algorithms_type2.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="468" data-original-width="247" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKBpfKdztdgpLCpxQUa14dzGvHbUBB5r_VBi7PkXUCk0ZwkuvMqlsJV7dffa_Fj8woM3fq5ciz-GPOBgPdXtvuNp0Va2LAI7w4KsTBDK_FHhfl0clpaM8hcJYJBXE8H5GaHxNY9HC1afSW/s320/Algorithms_type2.gif" width="168" /></a></div>
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With practice, students are able to estimate closer to the product they are trying to reach, and they can eventually practice long division as traditionally taught, but while keeping the value of the unit in mind and not needing mnemonics. Teachers can assess understanding by asking the student to explain his or her work using place value language.</div>
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The traditional algorithm is usually considered most efficient. However, for students with a command of multiplication, short division is even more efficient. An example is shown here. Again, teachers can assess understanding by requesting an explanation of the thinking involved.<br />
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<em style="background-color: white; color: #202020; font-family: Helvetica; font-size: 16px; text-align: start;"><span style="font-size: 12px;">Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at<span class="Apple-converted-space"> </span><a href="http://susanmidlarsky.com/" style="color: #2baadf; text-decoration-line: none;">susanmidlarsky.com</a>.</span></em><br />
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-28393976283512509262018-11-06T19:42:00.000-08:002018-11-06T19:42:19.612-08:00Problem Solving Task<h1 style="color: #202020; font-family: Helvetica; font-size: 26px; line-height: 32.5px; margin: 0px; padding: 0px; text-align: center; text-size-adjust: auto;">
Problem Solving Task<br /></h1>
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Jennifer, Rory, and Wayne were all born two years apart. Jennifer is the oldest, and Wayne is the youngest. The sum of their ages is 39. How old is Rory?</div>
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Solution and Discussion: </h3>
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This problem is similar to “consecutive integer” problems commonly presented in pre-algebra. Those who are comfortable with variables would likely solve for a variable. However, with Rory being the middle child, a common mistake would be to forget to add 2 to the youngest age to find Rory’s age.</div>
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Here is a sample bar model solution. It supports visualization of the situation and the difference in ages. It also supports algebraic thinking and attention to the variable for which the student is solving in the end. Practice in this type of approach leads to fewer mistakes of the type mentioned earlier. <a href="https://www.matholia.com/uscc/apps/tools/mt_f4726_kqtjz?cid=601">The Matholia Bar Model</a> tool can help students develop these visualizations.</div>
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<br />Bar Model Solution:</h3>
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Algebraic Solution:</h3>
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-51758507170332906392018-10-23T20:20:00.002-07:002018-10-23T20:33:32.126-07:00The Importance of the Unit: Part I<div style="line-height: 24px; margin-bottom: 10px; margin-top: 10px; padding: 0px; text-align: justify;">
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<span style="color: #202020; font-family: "helvetica";">An area of struggle for students is manipulating numbers once they get beyond basic counting numbers. A big reason for this struggle is that when manipulating multiple digits, students get intimidated by the larger numbers. They are also confused by how various digits work in algorithms. This shows up with operations on multiple-digit numbers, fractions, and decimals.</span><br />
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<span style="color: #202020; font-family: "helvetica";">One early source of the challenges may be the way we count. In many Latin-based languages, the teen numbers follow their own naming rules and don’t obey anything resembling logic. For example, what on earth does “eleven” mean? And what about “sixteen” -- why is the “teen” or tens named after the ones?</span><br />
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<span style="color: #202020; font-family: "helvetica";">Counting returns to normal once you get to the twenties, with twenty-one, twenty-two, etc. following place value order and the way we write them. But those teen numbers confuse young brains and can be responsible for many of the errors young children make, such as writing “seventeen” as 71 and not 17. They are being logical, generalizing the rules for writing numbers based on language; it’s our language that is illogical.</span><br />
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<span style="color: #202020; font-family: "helvetica";">Contrast this to languages such as those spoken in China, Japan, and South Korea, in which counting proceeds according to place value. They count 1-10 with individual words, like in English, but subsequent numbers are represented by place value: ten-one, ten-two, ...., ten-nine, two-ten, two-ten-one, two-ten-two, etc.</span><br />
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<span style="color: #202020; font-family: "helvetica";">This language advantage leads to a mathematical advantage. Language researchers believe this is the reason children from those countries can, by the end of preschool, count to 100 consistently, and solve almost three times as many simple arithmetic problems as US children. This gap can be closed by teaching the “say ten” way of counting, mirroring the counting used in these Southeast Asian countries. For example, children would learn to count the usual way, but then to rename the counting sequence using Say Ten counting.</span><br />
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<span style="color: #202020; font-family: "helvetica";">In conjunction with working with manipulatives to show that every number is a sum of ones, and that a ten is a shorter way of naming ten ones (<i>renaming</i>), students can develop a strong sense of place value that makes numbers easier to work with. They learn that digits work similarly, but it’s the place that gives the digit its value. Using the <b>Matholia place value disks tool</b> and <b>place value strips tool</b> can help the student see these dynamics in action.</span></div>
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<span style="color: #202020; font-family: "helvetica";">Students can also use the 10-row Rekenrek to track the tens and ones as they count. This will reinforce the concept of the “make-ten” strategy for addition and subtraction, which then can be generalized to multi-digit numbers. For example, adding 12 to 29, a student can decompose either number to make a ten and then add the remainder. A visual example of the mental math, using number bonds, is shown below. These strategies are practiced extensively in Primary 2.</span><span style="color: #202020; font-family: "helvetica";"><br /></span><br />
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<span style="color: #202020; font-family: helvetica;">Then if the student encounters 290 + 120, the same strategies can be used, just renaming it as “29 tens plus 12 tens,” understanding that the sum will be in tens, so 41 tens or 410.</span><br />
<span style="color: #202020; font-family: "helvetica";">The ways this understanding extends to fractions and decimals will be addressed in future articles.</span></div>
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<em><span style="font-size: 12px;">Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at<span class="Apple-converted-space"> </span><a href="http://susanmidlarsky.com/" style="color: #2baadf;">susanmidlarsky.com</a>.</span></em><br />
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<em>The Matholia Team</em></div>
Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-50945376629767929932018-10-16T20:42:00.000-07:002018-10-16T20:42:50.261-07:00Bar Modelling For Two Types of Division
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Many people aren’t aware that there are two different types of division. Even if you search the web, only one type mostly comes up. It’s the type we learn first: that when we share one quantity, we split it into equal parts. This is <i>partitive</i> division, or dividing a quantity into parts. When we divide a whole amount, partitive division tells us how many items there are in each group. Let’s look at a couple of examples.</div>
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<i>Jeanine bought 24 buns for a party. She wanted to put an equal amount on each of 6 tables.<span class="Apple-converted-space"> </span></i></div>
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<i>How many buns should she put on each table?</i></div>
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<i>Jeanine should put 4 buns on each table.</i></div>
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<i>Armando earned $56 for a day’s work.<span class="Apple-converted-space"> </span></i></div>
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<i>If he worked 7 hours, how much was his hourly wage?</i></div>
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<i>Armando’s hourly wage was $8.</i></div>
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The other type is <i>quotative</i> division, and it’s less familiar to most people. In quotative division, we find the number of groups, not how many in each group. It is also known as measurement division, or finding how many of a certain unit it will take to measure something. Let’s look at a couple of examples.</div>
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<i>When the party planner was setting up, he found he had 32 place cards. He would like to put four on each table. How many tables would he need to set up?</i></div>
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<i>The party planner would need to set up 8 tables.</i></div>
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<i>For a pizza party, a parent has a maximum budget of 6 pizzas. He plans on each child being allowed three slices. This pizza parlour slices each pizza into 8 slices. How many children can be invited to the party?</i></div>
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<i>16 children can be invited to the party.</i></div>
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The above example is as interesting way to model dividing a whole number by a fraction (an upper primary standard in most countries). Although bar modelling may not be as useful as a tool to calculate quotative division problems, it can be very helpful to help students visualize the problem.</div>
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Often when teaching such problems, many teachers resort to multiplying by the reciprocal, or 6 ×<span class="Apple-converted-space"> 8/3 </span>(We need to avoid teaching Keep-Change-Flip though, as this 3 mnemonic can lead to lots to misconceptions.).</div>
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Did you know that you can also divide using common denominators? You can</div>
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change 6 to 48/8<span class="Apple-converted-space"> </span>and solve<span class="Apple-converted-space"> 48/8 </span>÷<span class="Apple-converted-space"> 3/8 </span>. See if you can figure out why this works.</div>
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Leave a comment with your thoughts!</div>
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<span style="font-family: Arial, Helvetica, sans-serif; font-size: xx-small;"><i>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at <a href="http://susanmidlarsky.com/" style="color: #888888; text-decoration-line: none;">susanmidlarsky.com</a>.</i></span></div>
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-5617847771076707902018-10-08T18:23:00.000-07:002018-10-08T18:23:26.800-07:00What is more important than the right answer in primary mathematics?<div class="MsoNormal" style="font-family: Cambria; margin: 0cm 0cm 0.0001pt;">
<span style="font-family: Arial;">“There is no right or wrong in mathematics:” a quote from a memoir by the daughter of Fischer Black, a famed US mathematician, that she remembers him saying often. What does that mean?</span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">When we learn how numbers work, we can do so in one of several ways. The way many of us have been brought up to learn mathematics is through memorization and learning procedures, such as algorithms. One example is the long division algorithm, which is opaque to most students and is arguably one of the most difficult to learn. Teachers and parents alike can relate to frustration as students confuse which number to put above the little house, which below, and more. </span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">Another issue with this algorithm is that the way most people learn it. The digits are independent of value. For example, when dividing 125 by 5, first you would see how many times 5 goes into 1, then 12, then… but 1 what? 12 what? If it were truly 1 or 12, wouldn’t the digits be written in different places?</span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">Disconnecting the procedure from the meaning behind it leads to magical thinking about numbers: that it doesn’t matter how or why it works, it just does. </span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">Another place this shows up is in multiplying or dividing by powers of ten. How many of us think of “moving the decimal” or “adding a zero”? Both of these are not only how mathematics <i>doesn’t</i>work, they lead to significant mathematical errors and lack of understanding in students. </span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">To promote numeracy, or paying the same amount of attention to all students being as successful and capable in mathematics as they are in reading and writing, the focus needs to shift away from whether or not the answer is correct and more towards the way people think about the mathematics. This involves developing the concept so the student can apply whatever problem solving procedure works for him/her. This also means exposure to a variety of representations of the mathematics, to improve flexibility in thinking. The model of teaching mathematics in Singapore, and the US Common Core standards, both push for that type of learning. This way, students learn how mathematics works, not just how to get the right answer.</span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">For example, ten years ago, when working with parents or teachers, I would ask the question, “What is Pi?” The answers would range from something to do with a circle, to 3.14, to “I don’t know.”</span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">These days, after years of celebrating Pi Day in schools in March, the teachers I meet are much more likely to say, “It’s a ratio,” or “It’s the ratio between the circumference of a circle and its diameter.” This is a much more conceptual definition that allows flexibility in thinking and the ability to recreate the formulas for the different measurements of a circle. </span><span style="font-family: -webkit-standard, serif;"><o:p></o:p></span></div>
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<span style="font-family: Arial;">If teachers or parents are working with students on mathematics, why not try shifting focus from the answer to how the students are thinking? Start by giving feedback, rather than “right” or “wrong,” that focuses a student on their thinking and allows the student to identify if they made a mistake in understanding how the numbers work. This should lead to greater competence and </span><span style="font-family: Arial, Helvetica, sans-serif;">confidence in their own ability to learn.</span></div>
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<span style="font-family: Arial, Helvetica, sans-serif; font-size: x-small;"><i>Susan Midlarsky is a Math Consultant, a Curriculum Writer, and is keenly interested in questions related to learning math. You can find out more about her at <a href="http://susanmidlarsky.com/">susanmidlarsky.com</a>.</i></span></div>
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<span style="font-family: Arial, Helvetica, sans-serif;">Related article:</span></div>
<b><a href="https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=3&cad=rja&uact=8&ved=2ahUKEwj_quakmPjdAhXPdN4KHZDvAI0QFjACegQIBxAB&url=https%3A%2F%2Fcitinewsroom.com%2F2018%2F10%2F02%2Fsingapore-abolishes-school-exam-rankings-says-learning-is-not-competition%2F&usg=AOvVaw1-9oX950xWHXT94cWwyvpe" target="_blank"><span style="font-family: Arial, Helvetica, sans-serif;">Singapore abolishes school exam rankings, says learning is not competition</span></a></b><br />
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<br />Matholia Mathematicshttp://www.blogger.com/profile/12907740086274170195noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-3998298849217293522018-02-19T20:01:00.000-08:002018-02-19T20:01:09.097-08:00Matholia Squares of the week!<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQbrtg6PXpxuvUCmy_qzbn4g0NyUJEGErnVrPbsoWUc-gnasAL5HK4d9ysnRIKdSJW6G-iOsJiBgoGmmxQvvhdlBnXJceaqFF_Jr047GTQwvL6xRgXwFb1d6ajdzEeAzX6GC8TTqAmMeFG/s1600/Level+1-2+matholia+squares+word+problems+ALL+TOPICS+COMBINED2.jpg" imageanchor="1"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQbrtg6PXpxuvUCmy_qzbn4g0NyUJEGErnVrPbsoWUc-gnasAL5HK4d9ysnRIKdSJW6G-iOsJiBgoGmmxQvvhdlBnXJceaqFF_Jr047GTQwvL6xRgXwFb1d6ajdzEeAzX6GC8TTqAmMeFG/s320/Level+1-2+matholia+squares+word+problems+ALL+TOPICS+COMBINED2.jpg" width="320" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFZxpNUPu_ShtLDlngW_TgkEvVpIrLfnVnRAeLx4iZKSEuB2HDe9Lpw52uh-66rryp3Mv1UUtPfGJQYWzceEkyfn6VJCoWDMnvwErFL4zrjhGYFwOhe6srWKZMANQpA84ZioPG4d1MqHca/s1600/Level+1-2+matholia+squares+word+problems+ALL+TOPICS+COMBINED.jpg" imageanchor="1"><img border="0" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFZxpNUPu_ShtLDlngW_TgkEvVpIrLfnVnRAeLx4iZKSEuB2HDe9Lpw52uh-66rryp3Mv1UUtPfGJQYWzceEkyfn6VJCoWDMnvwErFL4zrjhGYFwOhe6srWKZMANQpA84ZioPG4d1MqHca/s320/Level+1-2+matholia+squares+word+problems+ALL+TOPICS+COMBINED.jpg" width="320" /></a>Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-2787508507360367972017-11-13T18:28:00.000-08:002017-11-13T18:56:57.277-08:00Matholia Squares of the week!<div class="separator" style="background-color: white; clear: both; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px;">
<span style="border: 0px; font-family: , , "segoe ui" , "roboto" , "helvetica" , "arial" , sans-serif; font-size: 14px; font-stretch: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Click on the link to see the answer! 😄</span></div>
<span style="border: 0px; font-family: , , "segoe ui" , "roboto" , "helvetica" , "arial" , sans-serif; font-size: 14px; font-stretch: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="https://www.youtube.com/watch?v=WGq0W_o-4ws&feature=youtu.be" style="color: #888888; text-decoration-line: none;">https://www.youtube.com/watch?v=WGq0W_o-4ws&feature=youtu.be</a></span><br />
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-74871337476587869402017-11-05T19:48:00.003-08:002017-11-05T19:53:22.701-08:00Matholia Squares of the Week!<br />
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Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-38835271410018023502017-10-29T20:06:00.000-07:002017-10-29T20:06:13.894-07:00Matholia Squares of the week!<div class="separator" style="clear: both; text-align: left;">
<br style="background-color: white; font-family: -apple-system, system-ui, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 14px;" /><span style="background-color: white; border: 0px; font-family: -apple-system, system-ui, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;">Click on the link to see the answer! 😄</span><br style="background-color: white; font-family: -apple-system, system-ui, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 14px;" /><span style="background-color: white; border: 0px; font-family: -apple-system, system-ui, "Segoe UI", Roboto, Helvetica, Arial, sans-serif; font-size: 14px; font-stretch: inherit; font-variant-numeric: inherit; line-height: inherit; margin: 0px; padding: 0px; vertical-align: baseline;"><a href="https://www.youtube.com/watch?v=7qD7-MfJSfY">https://www.youtube.com/watch?v=7qD7-MfJSfY</a></span></div>
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<br />Matholiahttp://www.blogger.com/profile/01167482109345870923noreply@blogger.com0tag:blogger.com,1999:blog-3770419529389105061.post-14761535066280792912017-10-22T19:56:00.001-07:002017-10-22T19:56:27.302-07:00Matholia squares of the week!<div class="separator" style="clear: both; text-align: center;">
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