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Using Manipulatives - TeacherVision
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</script> <!-- End VWO Async SmartCode --> <meta charset="utf-8" /> <title>Using Manipulatives - TeacherVision</title> <script>(function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0];var j=d.createElement(s);var dl=l!='dataLayer'?'&l='+l:'';j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl+'';j.async=true;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-TLN2FX');</script> <script>window.dataLayer = window.dataLayer || []; window.dataLayer.push([{'pageType' : 'TeachingResource','userState' : 'logged-out','plan' : 'No Plan','subStatus' : 'anonymous','vertical' : '(TV)Subjects','level_2' : '(TV)Teaching Strategies','level_3' : '(TV)Tools and Instruments','level_4' : '(TV)Mathematics','level_5' : '(TV)Professional Development','sku' : 'Pearson Achievement Solutions','publisher' : 'Pearson Achievement Solutions'}]);</script> <meta property="fb:app_id" content="1090629627653802" /> <meta name="referrer" content="origin" /> <meta property="og:site_name" content="TeacherVision" /> <meta property="og:type" content="Article" /> <meta property="og:url" content="https://www.teachervision.com/professional-development/using-manipulatives" /> <meta property="og:title" content="Using Manipulatives" /> <meta property="og:description" content="Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics. They can be used to introduce, practice, or remediate a concept." /> <meta property="og:image" content="https://www.teachervision.com/themes/teachervision/images/og_logo_pos.jpg" /> <meta property="og:updated_time" content="2007-03-15T15:17:23-04:00" /> <meta name="Generator" content="Drupal 10 (https://www.drupal.org)" /> <meta name="MobileOptimized" content="width" /> <meta name="HandheldFriendly" content="true" /> <meta name="viewport" content="width=device-width, initial-scale=1.0" /> <script type="application/ld+json">{ "@context": "https://schema.org", "@graph": [ { "@type": "Article", "headline": "Using Manipulatives", "name": "Using Manipulatives", "description": "Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics. They can be used to introduce, practice, or remediate a concept.", "image": { "@type": "ImageObject", "representativeOfPage": "True", "url": "https://www.teachervision.com/themes/teachervision/images/logo_pos.png", "width": "175", "height": "226" }, "datePublished": "2007-03-15T15:17:23-0400", "dateModified": "2019-11-15T01:12:57-0500", "author": { "@type": "Person", "name": "TeacherVision Staff", "url": "https://www.teachervision.com/advisory-board/teachervision-staff" }, "publisher": { "@type": "Organization", "name": "TeacherVision", "logo": { "@type": "ImageObject", "url": "https://www.teachervision.com/themes/teachervision/images/logo_pos.png", "width": "280", "height": "100" } }, "mainEntityOfPage": "https://www.teachervision.com/professional-development/using-manipulatives", "articleBody": "Using Manipulatives What Is It? Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics. They can be used to introduce, practice, or remediate a concept. A manipulative may be as simple as grains of rice or as sophisticated as a model of our solar system. They may be store-bought, brought from home, or teacher- or student-made. They can be used in all areas of math instruction-teaching number and operations, algebra, geometry, measurement, and so on-and across all grade levels. The following are a few manipulatives that might be used at the elementary level. Number and Operations Counters can be used to teach one-to-one correspondence, ordinal numbers, and basic addition and subtraction. Two-sided counters can be used to model one-to-one correspondence, addition and subtraction, or skip-count. Place-value mats can be used to show each digit\u0027s place value when using base-10 blocks. Base-10 blocks can be used to model the variety of ways a number can be represented, and to model regrouping when adding, subtracting, multiplying, and dividing. Money, in coins or bills, can be used to count, skip-count, or model regrouping when adding or subtracting. Fraction strips can be used to show equivalent fractions, to add fractions, to subtract fractions, or to find common denominators. Algebra Pattern blocks can be used to create, find, or extend patterns. Attribute blocks can be used to sort and classify according to shape, color, size, or other attributes. Pan balances or scales can be used with objects or weights to show equivalent values. Geometry Geoboards can be used to identify simple geometric shapes and describe their properties and to develop spatial sense. Geometric-solid models can be used to teach nets or spatial reasoning. Measurement Standard and non-standard rulers and measuring cups can be used to represent length or volume. Tiles can be used to find the area or the perimeter of an object. Data Analysis and Probability Spinners can be used to find the experimental probability of landing on a designated area. Number cubes or dice can be used to find the experimental probability of rolling a certain number or a combination of numbers. Why Is It Important? According to learning theory based on psychologist Jean Piaget\u0027s research, children are active learners who master concepts by progressing through three levels of knowledge--concrete, pictorial, and abstract. The use of manipulatives enables students to explore concepts at the first, or concrete, level of understanding. When students manipulate objects, they are taking the necessary first steps toward building understanding and internalizing math processes and procedures. For example, when learning to add fractions, students can use fraction strips to represent each addend. They then add the fractional parts to find the sum. After practicing with these, they can progress to finding sums for problems on paper, represented by pictures with corresponding fraction numerals (pictorial level). Over time, they will devise strategies and apply algorithms so they can find sums when given only the addition expression (abstract level). For a learner to understand and handle a concept a the abstract or symbolic level successfully, he or she must first understand the concept at the other two levels, in the order given-concrete then pictorial. Using concrete manipulatives is the first step to using mental images and models. When students demonstrate understanding with the concept at this physical, or concrete, level then they are ready to move to the next level, where they can apply their knowledge using representations of the objects in place of the objects themselves. Young children often solve addition and subtraction problems by counting concrete objects, beginning with their fingers. They go on to use concrete objects such as base-ten blocks and counters to develop more sophisticated problem-solving strategies based on what they know about counting (Siegler 1996). Concrete models can help students represent numbers and develop number sense; they can also help bring meaning to written symbols and can be useful in building place-value concepts. You should try to \u0022get inside students\u0027 heads\u0022 as they work with concrete materials by asking questions that elicit their thinking and reasoning. In this way, you can get a better sense of what students know and don\u0027t know, along with identifying misconceptions, thereby developing a basis for intervention strategies. You should also choose interesting tasks using manipulatives (such as an abacus) that engage students in mathematical thinking and reasoning, which builds their understanding of numbers as well as relationships among numbers (NCTM 2000). In fact, research shows that using manipulatives can contribute to the development of well-grounded, interconnected understandings of mathematical ideas. Students can more easily remember what they did and explain what they were thinking when they used manipulatives to solve a problem (Stein 2001)." } ] }</script> <style>body{font-family:'Sofia Sans',system-ui,sans-serif;font-weight:400}.hidden{display:none}.visually-hidden{position:absolute!important;overflow:hidden;clip:rect(1px,1px,1px,1px);width:1px;height:1px;word-wrap:normal}form[id^=favorites-addcollection] details summary::-webkit-details-marker{display:none}form[id^=favorites-addcollection]:not(.create-folder-form) .add-collection,form[id^=favorites-addcollection]:not(.create-folder-form) .form-item-cname{display:none}form[id^=favorites-addcollection]:not(.create-folder-form) .open-create-folder{display:block}.print-container-wrapper form[id^=favorites-addcollection]>details+.details-wrapper{border:1px solid #adb5bd;border-radius:1rem;bottom:auto!important;box-shadow:0 .5rem 1rem rgba(0,0,0,.15);display:block!important;max-height:0;max-width:calc(100vw - 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var(--bs-table-bg-state,var(--bs-table-bg-type,var(--bs-table-accent-bg)));color:var(--bs-table-color-state,var(--bs-table-color-type,var(--bs-table-color)));padding:.5rem}.table>tbody{vertical-align:inherit}.table>thead{vertical-align:bottom}.form-control{-webkit-appearance:none;-moz-appearance:none;appearance:none;background-clip:padding-box;background-color:var(--bs-body-bg);border:var(--bs-border-width) solid var(--bs-border-color);border-radius:var(--bs-border-radius);color:var(--bs-body-color);display:block;font-size:1rem;font-weight:400;line-height:1.5;padding:.375rem .75rem;width:100%}.form-control::-webkit-date-and-time-value{height:1.5em;margin:0;min-width:85px}.form-control::-webkit-datetime-edit{display:block;padding:0}.form-control::-webkit-input-placeholder{color:var(--bs-secondary-color);opacity:1}.form-control::-moz-placeholder{color:var(--bs-secondary-color);opacity:1}.form-control::-ms-input-placeholder{color:var(--bs-secondary-color);opacity:1}.form-control::-webkit-file-upload-button{-webkit-margin-end:.75rem;background-color:var(--bs-tertiary-bg);border:0 solid;border-color:inherit;border-inline-end-width:var(--bs-border-width);border-radius:0;color:var(--bs-body-color);margin:-.375rem -.75rem;margin-inline-end:.75rem;padding:.375rem .75rem}.form-check{display:block;margin-bottom:0;min-height:1.5rem;padding-left:0}.form-check .form-check-input{float:left;margin-left:0}.form-check-input{--bs-form-check-bg:var(--bs-body-bg);-webkit-appearance:none;-moz-appearance:none;appearance:none;background-color:var(--bs-form-check-bg);background-image:var(--bs-form-check-bg-image);background-position:50%;background-repeat:no-repeat;background-size:contain;border:var(--bs-border-width) solid var(--bs-border-color);flex-shrink:0;height:1.5rem;margin-top:0;print-color-adjust:exact;vertical-align:top;width:1.5rem}.form-check-input[type=checkbox]{border-radius:.25em}.dropdown{position:relative}.dropdown-toggle{white-space:nowrap}.dropdown-toggle:after{border-bottom:0;border-left:.3em solid transparent;border-right:.3em solid transparent;border-top:.3em solid;content:"";display:inline-block;margin-left:.255em;vertical-align:.255em}.dropdown-menu{--bs-dropdown-zindex:1000;--bs-dropdown-min-width:10rem;--bs-dropdown-padding-x:0;--bs-dropdown-padding-y:0.5rem;--bs-dropdown-spacer:0.125rem;--bs-dropdown-font-size:1rem;--bs-dropdown-color:var(--bs-body-color);--bs-dropdown-bg:var(--bs-body-bg);--bs-dropdown-border-color:var(--bs-border-color-translucent);--bs-dropdown-border-radius:var(--bs-border-radius);--bs-dropdown-border-width:var(--bs-border-width);--bs-dropdown-inner-border-radius:calc(var(--bs-border-radius) - var(--bs-border-width));--bs-dropdown-divider-bg:var(--bs-border-color-translucent);--bs-dropdown-divider-margin-y:0.5rem;--bs-dropdown-box-shadow:var(--bs-box-shadow);--bs-dropdown-link-color:#1e3960;--bs-dropdown-link-hover-color:#1e3960;--bs-dropdown-link-hover-bg:none;--bs-dropdown-link-active-color:#fff;--bs-dropdown-link-active-bg:#1e3960;--bs-dropdown-link-disabled-color:var(--bs-tertiary-color);--bs-dropdown-item-padding-x:1rem;--bs-dropdown-item-padding-y:0.25rem;--bs-dropdown-header-color:#6c757d;--bs-dropdown-header-padding-x:1rem;--bs-dropdown-header-padding-y:0.5rem;background-clip:padding-box;background-color:var(--bs-dropdown-bg);border:var(--bs-dropdown-border-width) solid var(--bs-dropdown-border-color);border-radius:var(--bs-dropdown-border-radius);color:var(--bs-dropdown-color);display:none;font-size:var(--bs-dropdown-font-size);list-style:none;margin:0;min-width:var(--bs-dropdown-min-width);padding:var(--bs-dropdown-padding-y) var(--bs-dropdown-padding-x);position:absolute;text-align:left;z-index:var(--bs-dropdown-zindex)}.dropdown-item{background-color:transparent;border:0;border-radius:var(--bs-dropdown-item-border-radius,0);clear:both;color:var(--bs-dropdown-link-color);display:block;font-weight:400;padding:var(--bs-dropdown-item-padding-y) var(--bs-dropdown-item-padding-x);text-align:inherit;white-space:nowrap;width:100%}.nav{--bs-nav-link-padding-x:1rem;--bs-nav-link-padding-y:0.5rem;--bs-nav-link-font-weight: ;--bs-nav-link-color:#1e3960;--bs-nav-link-hover-color:#172b48;--bs-nav-link-disabled-color:var(--bs-secondary-color);display:flex;flex-wrap:wrap;list-style:none;margin-bottom:0;padding-left:0}.nav-link{background:0 0;border:0;color:var(--bs-nav-link-color);display:block;font-size:var(--bs-nav-link-font-size);font-weight:var(--bs-nav-link-font-weight);padding:var(--bs-nav-link-padding-y) var(--bs-nav-link-padding-x)}.navbar{--bs-navbar-padding-x:0;--bs-navbar-padding-y:0.5rem;--bs-navbar-color:rgba(var(--bs-emphasis-color-rgb),0.65);--bs-navbar-hover-color:rgba(var(--bs-emphasis-color-rgb),0.8);--bs-navbar-disabled-color:rgba(var(--bs-emphasis-color-rgb),0.3);--bs-navbar-active-color:rgba(var(--bs-emphasis-color-rgb),1);--bs-navbar-brand-padding-y:0.3125rem;--bs-navbar-brand-margin-end:none;--bs-navbar-brand-font-size:1.25rem;--bs-navbar-brand-color:rgba(var(--bs-emphasis-color-rgb),1);--bs-navbar-brand-hover-color:rgba(var(--bs-emphasis-color-rgb),1);--bs-navbar-nav-link-padding-x:0.5rem;--bs-navbar-toggler-padding-y:0.75rem;--bs-navbar-toggler-padding-x:1rem;--bs-navbar-toggler-font-size:1.25rem;--bs-navbar-toggler-border-color:rgba(var(--bs-emphasis-color-rgb),0.15);--bs-navbar-toggler-border-radius:1rem;--bs-navbar-toggler-focus-width:0.25rem;align-items:center;display:flex;flex-wrap:wrap;justify-content:space-between;padding:var(--bs-navbar-padding-y) var(--bs-navbar-padding-x);position:relative}.navbar>.container{align-items:center;display:flex;flex-wrap:inherit;justify-content:space-between}.navbar-brand{color:var(--bs-navbar-brand-color);font-size:var(--bs-navbar-brand-font-size);margin-right:var(--bs-navbar-brand-margin-end);padding-bottom:var(--bs-navbar-brand-padding-y);padding-top:var(--bs-navbar-brand-padding-y);white-space:nowrap}.navbar-toggler{background-color:transparent;border:var(--bs-border-width) solid var(--bs-navbar-toggler-border-color);border-radius:var(--bs-navbar-toggler-border-radius);color:var(--bs-navbar-color);font-size:var(--bs-navbar-toggler-font-size);line-height:1;padding:var(--bs-navbar-toggler-padding-y) var(--bs-navbar-toggler-padding-x)}.breadcrumb{--bs-breadcrumb-padding-x:0;--bs-breadcrumb-padding-y:0;--bs-breadcrumb-margin-bottom:1rem;--bs-breadcrumb-bg: ;--bs-breadcrumb-border-radius: ;--bs-breadcrumb-divider-color:#73a6ff;--bs-breadcrumb-item-padding-x:0.5rem;--bs-breadcrumb-item-active-color:var(--bs-secondary-color);background-color:var(--bs-breadcrumb-bg);border-radius:var(--bs-breadcrumb-border-radius);display:flex;flex-wrap:wrap;font-size:var(--bs-breadcrumb-font-size);list-style:none;margin-bottom:var(--bs-breadcrumb-margin-bottom);padding:var(--bs-breadcrumb-padding-y) var(--bs-breadcrumb-padding-x)}.breadcrumb-item.active{color:var(--bs-breadcrumb-item-active-color)}.btn-close{--bs-btn-close-color:#000;--bs-btn-close-opacity:0.5;--bs-btn-close-hover-opacity:0.75;--bs-btn-close-focus-shadow:0 0 0 0.25rem rgba(219,0,18,.25);--bs-btn-close-focus-opacity:1;--bs-btn-close-disabled-opacity:0.25;--bs-btn-close-white-filter:invert(1) grayscale(100%) brightness(200%);background:transparent var(--bs-btn-close-bg) center/1em auto no-repeat;border:0;border-radius:.375rem;box-sizing:content-box;height:1em;opacity:var(--bs-btn-close-opacity);padding:.25em;width:1em}.btn-close{color:var(--bs-btn-close-color)}.offcanvas{--bs-offcanvas-zindex:1045;--bs-offcanvas-width:420px;--bs-offcanvas-height:30vh;--bs-offcanvas-padding-x:1rem;--bs-offcanvas-padding-y:1rem;--bs-offcanvas-color:var(--bs-body-color);--bs-offcanvas-bg:var(--bs-body-bg);--bs-offcanvas-border-width:var(--bs-border-width);--bs-offcanvas-border-color:var(--bs-border-color-translucent);--bs-offcanvas-box-shadow:var(--bs-box-shadow-sm);--bs-offcanvas-title-line-height:1.5}.offcanvas{background-clip:padding-box;background-color:var(--bs-offcanvas-bg);bottom:0;color:var(--bs-offcanvas-color);display:flex;flex-direction:column;max-width:100%;outline:0;position:fixed;visibility:hidden;z-index:var(--bs-offcanvas-zindex)}.offcanvas.offcanvas-start{border-right:var(--bs-offcanvas-border-width) solid var(--bs-offcanvas-border-color);left:0;top:0;-webkit-transform:translateX(-100%);transform:translateX(-100%);width:var(--bs-offcanvas-width)}.offcanvas-header{align-items:center;display:flex;padding:var(--bs-offcanvas-padding-y) var(--bs-offcanvas-padding-x)}.offcanvas-header .btn-close{margin:calc(var(--bs-offcanvas-padding-y)*-.5) calc(var(--bs-offcanvas-padding-x)*-.5) calc(var(--bs-offcanvas-padding-y)*-.5) auto;padding:calc(var(--bs-offcanvas-padding-y)*.5) calc(var(--bs-offcanvas-padding-x)*.5)}.offcanvas-body{flex-grow:1;overflow-y:auto;padding:var(--bs-offcanvas-padding-y) var(--bs-offcanvas-padding-x)}.overflow-auto{overflow:auto!important}.overflow-x-hidden{overflow-x:hidden!important}.d-block{display:block!important}.d-flex{display:flex!important}.d-none{display:none!important}.position-relative{position:relative!important}.position-absolute{position:absolute!important}.position-fixed{position:fixed!important}.top-50{top:50%!important}.bottom-0{bottom:0!important}.start-0{left:0!important}.end-0{right:0!important}.translate-middle-y{-webkit-transform:translateY(-50%)!important;transform:translateY(-50%)!important}.border{border:var(--bs-border-width) var(--bs-border-style) var(--bs-border-color)!important}.border-0{border:0!important}.border-top{border-top:var(--bs-border-width) var(--bs-border-style) var(--bs-border-color)!important}.border-end-0{border-right:0!important}.border-bottom{border-bottom:var(--bs-border-width) var(--bs-border-style) var(--bs-border-color)!important}.border-start-0{border-left:0!important}.border-info{--bs-border-opacity:1;border-color:rgba(var(--bs-info-rgb),var(--bs-border-opacity))!important}.border-light-subtle{border-color:var(--bs-light-border-subtle)!important}.w-100{width:100%!important}.mw-100{max-width:100%!important}.vw-100{width:100vw!important}.h-100{height:100%!important}.flex-column{flex-direction:column!important}.flex-wrap{flex-wrap:wrap!important}.flex-nowrap{flex-wrap:nowrap!important}.justify-content-start{justify-content:flex-start!important}.justify-content-end{justify-content:flex-end!important}.justify-content-between{justify-content:space-between!important}.align-items-center{align-items:center!important}.m-0{margin:0!important}.mx-2{margin-left:.5rem!important;margin-right:.5rem!important}.my-3{margin-bottom:1rem!important;margin-top:1rem!important}.my-4{margin-bottom:1.5rem!important;margin-top:1.5rem!important}.mt-0{margin-top:0!important}.mt-2{margin-top:.5rem!important}.me-1{margin-right:.25rem!important}.me-2{margin-right:.5rem!important}.mb-4{margin-bottom:1.5rem!important}.ms-1{margin-left:.25rem!important}.p-0{padding:0!important}.p-3{padding:1rem!important}.p-4{padding:1.5rem!important}.px-4{padding-left:1.5rem!important;padding-right:1.5rem!important}.py-3{padding-bottom:1rem!important;padding-top:1rem!important}.py-4{padding-bottom:1.5rem!important;padding-top:1.5rem!important}.pt-0{padding-top:0!important}.pt-4{padding-top:1.5rem!important}.pe-3{padding-right:1rem!important}.pe-5{padding-right:3rem!important}.pb-5{padding-bottom:3rem!important}.gap-3{gap:1rem!important}.gap-4{gap:1.5rem!important}.row-gap-3{row-gap:1rem!important}.row-gap-4{row-gap:1.5rem!important}.column-gap-5{-webkit-column-gap:3rem!important;-moz-column-gap:3rem!important;column-gap:3rem!important}.fs-1{font-size:calc(1.375rem + 1.5vw)!important}.fs-3{font-size:calc(1.3rem + .6vw)!important}.fs-4{font-size:calc(1.275rem + .3vw)!important}.fs-5{font-size:1.2rem!important}.fw-semibold{font-weight:600!important}.fw-bold{font-weight:700!important}.text-center{text-align:center!important}.text-decoration-none{text-decoration:none!important}.text-wrap{white-space:normal!important}.text-primary{--bs-text-opacity:1;color:rgba(219,0,18,var(--bs-text-opacity))!important}.text-info{--bs-text-opacity:1;color:rgba(30,57,96,var(--bs-text-opacity))!important}.text-blue{--bs-text-opacity:1;color:rgba(5,97,255,var(--bs-text-opacity))!important}.bg-light{--bs-bg-opacity:1;background-color:rgba(243,244,248,var(--bs-bg-opacity))!important}.bg-white{--bs-bg-opacity:1;background-color:rgba(255,255,255,var(--bs-bg-opacity))!important}.rounded-0{border-radius:0!important}.rounded-pill{border-radius:var(--bs-border-radius-pill)!important}.rounded-bottom-4{border-bottom-left-radius:var(--bs-border-radius-xl)!important;border-bottom-right-radius:var(--bs-border-radius-xl)!important}.z-2{z-index:2!important}.z-3{z-index:3!important}@media (min-width:576px){.d-sm-block{display:block!important}.d-sm-none{display:none!important}.mx-sm-3{margin-left:1rem!important;margin-right:1rem!important}}@media (min-width:992px){.d-lg-block{display:block!important}.d-lg-flex{display:flex!important}.d-lg-none{display:none!important}.flex-lg-nowrap{flex-wrap:nowrap!important}.align-items-lg-end{align-items:flex-end!important}.pt-lg-3{padding-top:1rem!important}.pe-lg-5{padding-right:3rem!important}.gap-lg-5{gap:3rem!important}}@media (min-width:1200px){.fs-1{font-size:2.5rem!important}.fs-3{font-size:1.75rem!important}.fs-4{font-size:1.5rem!important}}.icon-link{align-items:center;-webkit-backface-visibility:hidden;backface-visibility:hidden;display:inline-flex;gap:.375rem;-webkit-text-decoration-color:rgba(var(--bs-link-color-rgb),var(--bs-link-opacity,.5));text-decoration-color:rgba(var(--bs-link-color-rgb),var(--bs-link-opacity,.5));text-underline-offset:.25em}.icon-link>.bi{fill:currentcolor;flex-shrink:0;height:1em;width:1em}.sticky-top{top:0}.sticky-top{position:-webkit-sticky;position:sticky;z-index:1020}.visually-hidden{clip:rect(0,0,0,0)!important;border:0!important;height:1px!important;margin:-1px!important;overflow:hidden!important;padding:0!important;white-space:nowrap!important;width:1px!important}.visually-hidden:not(caption){position:absolute!important}.btn{display:inline-block}.btn{--bs-btn-padding-x:1rem;--bs-btn-padding-y:0.5rem;--bs-btn-font-family: ;--bs-btn-font-size:1rem;--bs-btn-font-weight:700;--bs-btn-line-height:1.5;--bs-btn-color:var(--bs-body-color);--bs-btn-bg:transparent;--bs-btn-border-width:var(--bs-border-width);--bs-btn-border-color:transparent;--bs-btn-border-radius:0.375rem;--bs-btn-hover-border-color:transparent;--bs-btn-box-shadow:inset 0 1px 0 hsla(0,0%,100%,.15),0 1px 1px rgba(0,0,0,.075);--bs-btn-disabled-opacity:0.65;--bs-btn-focus-box-shadow:0 0 0 0.25rem rgba(var(--bs-btn-focus-shadow-rgb),.5);background-color:var(--bs-btn-bg);border:var(--bs-btn-border-width) solid var(--bs-btn-border-color);border-radius:var(--bs-btn-border-radius);color:var(--bs-btn-color);font-family:var(--bs-btn-font-family);font-size:var(--bs-btn-font-size);font-weight:var(--bs-btn-font-weight);line-height:var(--bs-btn-line-height);padding:var(--bs-btn-padding-y) var(--bs-btn-padding-x);text-align:center;vertical-align:middle}.btn-primary{--bs-btn-color:#fff;--bs-btn-bg:#db0012;--bs-btn-border-color:#db0012;--bs-btn-hover-color:#fff;--bs-btn-hover-bg:#a4000e;--bs-btn-hover-border-color:#af000e;--bs-btn-focus-shadow-rgb:224,38,54;--bs-btn-active-color:#fff;--bs-btn-active-bg:#af000e;--bs-btn-active-border-color:#a4000e;--bs-btn-active-shadow:inset 0 3px 5px rgba(0,0,0,.125);--bs-btn-disabled-color:#fff;--bs-btn-disabled-bg:#db0012;--bs-btn-disabled-border-color:#db0012}.btn-secondary{--bs-btn-color:#000;--bs-btn-bg:#fccd40;--bs-btn-border-color:#fccd40;--bs-btn-hover-color:#000;--bs-btn-hover-bg:#fdda70;--bs-btn-hover-border-color:#fcd253;--bs-btn-focus-shadow-rgb:214,174,54;--bs-btn-active-color:#000;--bs-btn-active-bg:#fdd766;--bs-btn-active-border-color:#fcd253;--bs-btn-active-shadow:inset 0 3px 5px rgba(0,0,0,.125);--bs-btn-disabled-color:#000;--bs-btn-disabled-bg:#fccd40;--bs-btn-disabled-border-color:#fccd40}.btn-info{--bs-btn-color:#fff;--bs-btn-bg:#1e3960;--bs-btn-border-color:#1e3960;--bs-btn-hover-color:#fff;--bs-btn-hover-bg:#172b48;--bs-btn-hover-border-color:#182e4d;--bs-btn-focus-shadow-rgb:64,87,120;--bs-btn-active-color:#fff;--bs-btn-active-bg:#182e4d;--bs-btn-active-border-color:#172b48;--bs-btn-active-shadow:inset 0 3px 5px rgba(0,0,0,.125);--bs-btn-disabled-color:#fff;--bs-btn-disabled-bg:#1e3960;--bs-btn-disabled-border-color:#1e3960}.btn-link{--bs-btn-font-weight:400;--bs-btn-color:var(--bs-link-color);--bs-btn-bg:transparent;--bs-btn-border-color:transparent;--bs-btn-hover-color:var(--bs-link-hover-color);--bs-btn-hover-border-color:transparent;--bs-btn-active-color:var(--bs-link-hover-color);--bs-btn-active-border-color:transparent;--bs-btn-disabled-color:#6c757d;--bs-btn-disabled-border-color:transparent;--bs-btn-box-shadow:0 0 0 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.form-item:not(.form-check){margin-bottom:1.25em;margin-top:1.25em;position:relative}#block-tv-radix-tvregistrationpopupblock{display:none}.dropdown-item,.dropdown-main-nav .col>a{font-weight:300}#main-nav .nav-link{border-bottom:3px solid transparent;padding:.5rem 0 0}#offcanvas-nav .dropdown-menu{position:static}#overlay{background:#000;bottom:0;display:none;left:0;opacity:.7;position:fixed;right:0;top:0;z-index:99998}#loading{left:50%;margin:-28px 0 0 -25px;top:50%}#loading,.admin-menu-icon{position:fixed;z-index:99999}.admin-menu-icon{display:none}:root{--bs-blue-light-rgb:115,166,255;--bs-blue-rgb:5,97,255;--bs-purple-light-rgb:193,142,224;--bs-purple-rgb:160,83,208;--bs-orange-rgb:246,142,66;--bs-orange-dark-rgb:190,84,9;--bs-yellow-dark-rgb:150,113,3;--bs-green-light-rgb:7,187,118;--bs-green-dark-rgb:6,127,80;--bs-cream-rgb:251,245,234;--bs-light-gray-rgb:173,181,189;--bs-dark-gray-rgb:73,80,87;--bs-infoplease-rgb:45,136,172}.oute_new_blog 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<i class="fa-solid fa-chevron-right text-blue fw-bold mx-2 mx-sm-3"></i> <li class="breadcrumb-item "> <a class="link-blue" href="/teaching-strategies">Teaching Resource</a> </li> <i class="fa-solid fa-chevron-right text-blue fw-bold mx-2 mx-sm-3"></i> <li class="breadcrumb-item "> <a class="link-blue" href="https://www.teachervision.com/subjects/mathematics">Mathematics</a> </li> <i class="fa-solid fa-chevron-right text-blue fw-bold mx-2 mx-sm-3"></i> <li class="breadcrumb-item active"> <span class="text-info fw-semibold">Using Manipulatives</span> </li> </ol> </nav> </div> <div id="block-tv-radix-page-title" class="block block--tv-radix-page-title"> <h1 class="page-title"> <span>Using Manipulatives</span> </h1> </div> </div> </header> <div class="container"> <div class="row"> <section id="main-content" class="page__content layout-content col-12 col-lg-9 pe-lg-5 ad-adjust"> <div id="block-tv-radix-content" class="block block--tv-radix-content block--system-main"> <article data-history-node-id="5239" class="node node--promoted node--type-articles articles--tv-full-node"> <div class="node__content"> <div class="field field--name-field-teacher-advisor field--type-entity-reference field--label-hidden field__item"> <article data-history-node-id="2319306" class="node node--type-author-profiles author-profiles--teacher-advisor-card"> <div class="field__items advisor-content-card d-flex gap-2 mb-2"> <div class="advisor-card-left-grid align-self-start rounded-circle overflow-hidden"> <div class="field field--name-field-advisor-profile-pic field--type-image field--label-hidden field__item"> <a href="https://www.teachervision.com/advisory-board/teachervision-staff" hreflang="en"> <picture> <source srcset="/sites/default/files/styles/scale50w_crop/public/2019-11/tv-apple.png.webp?itok=XBBP7Hc5 1x" media="all and (min-width: 0em)" type="image/webp"/> <img loading="lazy" src="https://www.teachervision.com/sites/default/files/styles/scale50w_crop/public/2019-11/tv-apple.png" width="50" height="50" alt="TeacherVision Staff" class="img-fluid"> </picture> </a> </div> </div> <div class="advisor-card-right-grid"> <div class="created-by">Authored by: <span class="created-by-anchor"><a href="https://www.teachervision.com/advisory-board/teachervision-staff"> <span>TeacherVision Staff</span> </a></span> </div> <div class="update-date">Last edited: November 15, 2019</div> </div> </div> </article> </div> <div class="field field--name-field-brief-description field--type-string-long field--label-hidden field__item"><h4>Learn how to effectively use fraction strips, spinners, counters, and more</h4> Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics. They can be used to introduce, practice, or remediate a concept. Use this resource to help your students learn how to use manipulatives successfully. <p>Looking for more manipulatives like <a href="/fractions/fraction-strips-grades-3-5">fraction strips</a>? Find them in our <a href="https://www.teachervision.com/subjects/mathematics">math resources center</a>.</p></div> <div class="asset-tagging row row-gap-2 mt-3"> <div class="col-sm-12 col-md-7"> <div class="field field--name-field-grade-level field--type-entity-reference field--label-above"> <div class="field__label">Grades:</div> <div class="field__items d-flex flex-wrap row-gap-1"> <div class="field__item px-2 border-end"><a href="/grades/pre-k">Pre-K</a> </div> <div class="field__item px-2 border-end"><a href="/grades/kindergarten">K</a> </div> <div class="field__item px-2 border-end"><a href="/grades/1st-grade">1</a> </div> <div class="field__item px-2 border-end"><a href="/grades/2nd-grade">2</a> </div> <div class="field__item px-2 border-end"><a href="/grades/3rd-grade">3</a> </div> <div class="field__item px-2 border-end"><a href="/grades/4th-grade">4</a> </div> <div class="field__item px-2 border-end"><a href="/grades/5th-grade">5</a> </div> <div class="field__item px-2 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class="field field--name-field-simplified-content-type field--type-entity-reference field--label-above"> <div class="field__label">Type:</div> <div class="field__items"> <div class="field__item"> <a href="/resources/teaching-resource">Teaching Resource</a> </div> </div> </div> <div class="field field--name-field-teaching-strategies field--type-entity-reference field--label-above"> <div class="field__label">Teaching Strategies:</div> <div class='field__items'> <div class="field__item"><a href="/teaching-strategies/professional-development" hreflang="en">Professional Development</a></div> </div> </div> </div> </div> <div class="print-container-wrapper d-flex flex-wrap align-items-center column-gap-5 row-gap-4 bg-light p-4 my-4"> <div id="favorites-section-5239" class="position-relative"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5239 needs-validation form" data-drupal-selector="favorites-addcollection" 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aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection" type="hidden" name="form_id" value="favorites_addcollection" class="form-control" aria-label="input"> </form> <div id="favoritesResponse-5239" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"> </div> </div> </div> <span class="print-page-link fs-5"> <a href="#" class="print-page link-dark" id="print_resource" data-print="login"> <i class="fa-solid fa-print me-1" aria-hidden="true"></i> Print </a> </span> <div id="custom-social-icons"> <ul class="share-buttons list-unstyled d-flex flex-wrap gap-3 fs-4 m-0"> <li><a href="https://www.pinterest.com/pin/create/button/?url=&description=" rel="nofollow noopener" target="_blank" data-pin-do="buttonPin" data-pin-custom="true" class="pinterest link-dark"><i class="fa-brands fa-pinterest"></i></a></li> <li><a role="button" title="Share this page to facebook" class="facebook link-dark" rel="nofollow noopener" target="_blank"><i class="fa-brands fa-square-facebook"></i></a></li> <li><a role="button" title="Share this page to twitter" class="twitter link-dark" rel="nofollow noopener" target="_blank"><i class="fa-brands fa-square-x-twitter"></i></a></li> <li><a role="button" title="Share this page to an email" class="email link-dark" rel="nofollow noopener" target="_blank"> <i class="fa-solid fa-envelope"></i></a></li> <li><a role="button" title="Google bookmark this page" class="gbookmark link-dark" rel="nofollow noopener" target="_blank"> <i class="fa-brands fa-google"></i></a></li> </ul> </div> <div class="addthis_inline_share_toolbox_4641"></div> </div> <div class="download-article my-4"> <a href="#" target="_blank" data-user-state="anonymous" class="btn btn-primary anonymous-download-pdf">Download</a></div> <div class="field field--name-body field--type-text-with-summary field--label-hidden field__item"><div class="top-page-nav">Page 1 of 4</div> <h2>Using Manipulatives</h2> <h3>What Is It?</h3> <p>Manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of <a href="/search">mathematics</a>. They can be used to introduce, practice, or remediate a concept. A manipulative may be as simple as grains of rice or as sophisticated as a model of our solar system. They may be store-bought, brought from home, or teacher- or student-made. They can be used in all areas of math instruction-teaching number and operations, algebra, geometry, measurement, and so on-and across all grade levels. </p> <p>The following are a few manipulatives that might be used at the elementary level. </p> <h4>Number and Operations</h4><ul><li><p><a href="/teaching-strategies/2-color-counters"><i>Counters</i></a> can be used to teach one-to-one correspondence, ordinal numbers, and basic addition and subtraction.</p></li> <li><p><i>Two-sided counters</i> can be used to model one-to-one correspondence, addition and subtraction, or skip-count.</p></li> <li><p><a href="/graph-chart-0/4-digit-place-value-charts"><i>Place-value mats</i></a> can be used to show each digit's place value when using base-10 blocks.</p></li> <li><p><i>Base-10 blocks</i> can be used to model the variety of ways a number can be represented, and to model regrouping when adding, subtracting, multiplying, and dividing.</p></li> <li><p><i>Money</i>, in <a href="/money/coins">coins</a> or <a href="/money/money-printable-dollar-bills">bills</a>, can be used to count, skip-count, or model regrouping when adding or subtracting.</p></li> <li><p><a href="/fractions/fraction-strips-grades-3-5"><i>Fraction strips</i></a> can be used to show equivalent fractions, to add fractions, to subtract fractions, or to find common denominators.</p></li></ul><h4>Algebra</h4><ul><li><p><a href="/geometry/pattern-blocks"><i>Pattern blocks</i></a> can be used to create, find, or extend patterns.</p></li> <li><p><a href="/geometry/attribute-shapes"><i>Attribute blocks</i></a> can be used to sort and classify according to shape, color, size, or other attributes.</p></li> <li><p><i>Pan balances or scales</i> can be used with objects or weights to show equivalent values. </p></li></ul><h4>Geometry</h4> <ul><li><p><a href="/tools-instruments/geoboard-dot-paper"><i>Geoboards</i></a> can be used to identify simple geometric shapes and describe their properties and to develop spatial sense.</p></li> <li><p><a href="/geometry/solids"><i>Geometric-solid models</i></a> can be used to teach nets or spatial reasoning.</p></li></ul><h4>Measurement</h4> <ul><li><p><i>Standard and non-standard <a href="/measurement/rulers">rulers</a> and <a href="/measurement/measuring-cups">measuring cups</a></i> can be used to represent length or volume.</p></li> <li><p><a href="/algebra/algebra-tiles"><i>Tiles</i></a> can be used to find the area or the perimeter of an object.</p></li></ul><h4>Data Analysis and Probability</h4> <ul><li><p><a href="/activity/number-spinners"><i>Spinners</i></a> can be used to find the experimental probability of landing on a designated area.</p></li> <li><p><i>Number cubes or dice</i> can be used to find the experimental probability of rolling a certain number or a combination of numbers.</p></li></ul><h3>Why Is It Important?</h3> <p>According to learning theory based on psychologist Jean Piaget's research, children are active learners who master concepts by progressing through three levels of knowledge--concrete, pictorial, and abstract. The use of manipulatives enables students to explore concepts at the first, or concrete, level of understanding. When students manipulate objects, they are taking the necessary first steps toward building understanding and internalizing math processes and procedures. For example, when learning to add <a href="/subjects/mathematics/decimals">fractions</a>, students can use <a href="/fractions/fraction-strips-grades-3-5">fraction strips</a> to represent each addend. They then add the fractional parts to find the sum. After practicing with these, they can progress to finding sums for problems on paper, represented by pictures with corresponding fraction numerals (pictorial level). Over time, they will devise strategies and apply algorithms so they can find sums when given only the addition expression (abstract level).</p> <p>For a learner to understand and handle a concept a the abstract or symbolic level successfully, he or she must first understand the concept at the other two levels, in the order given-concrete then pictorial. Using concrete manipulatives is the first step to using mental images and models. When students demonstrate understanding with the concept at this physical, or concrete, level then they are ready to move to the next level, where they can apply their knowledge using <i>representations</i> of the objects in place of the objects themselves.</p> <p>Young children often solve <a href="/subjects/mathematics/addition">addition and subtraction</a> problems by counting concrete objects, beginning with their fingers. They go on to use concrete objects such as base-ten blocks and counters to develop more sophisticated problem-solving strategies based on what they know about counting (Siegler 1996).</p> <p>Concrete models can help students represent numbers and develop number sense; they can also help bring meaning to written symbols and can be useful in building place-value concepts. You should try to "get inside students' heads" as they work with concrete materials by asking questions that elicit their thinking and reasoning. In this way, you can get a better sense of what students know and don't know, along with identifying misconceptions, thereby developing a basis for intervention strategies. You should also choose interesting tasks using manipulatives (such as an abacus) that engage students in mathematical thinking and reasoning, which builds their understanding of numbers as well as relationships among numbers (NCTM 2000).</p> <p>In fact, research shows that using manipulatives can contribute to the development of well-grounded, interconnected understandings of mathematical ideas. Students can more easily remember what they did and explain what they were thinking when they used manipulatives to solve a problem (Stein 2001).</p></div> </div> <!--<h2>In This Article:</h2>--> <ul class="inthis_articles custom-pagignation"> <li class="prev-itm">Prev</li> <li class="current-itm">1</li> <li><a href="https://www.teachervision.com/professional-development/using-manipulatives?page=2">2</a></li> <li><a href="https://www.teachervision.com/professional-development/using-manipulatives?page=3">3</a></li> <li><a href="https://www.teachervision.com/professional-development/using-manipulatives?page=4">4</a></li> <li><a href="https://www.teachervision.com/professional-development/using-manipulatives?page=2">Next</a></li> </ul> </article> </div> <div id="block-tv-radix-subscriptionmessageblock" class="block block--tv-radix-subscriptionmessageblock"> <div class="fw-bold"> </div> </div> <div class="views-element-container block block--tv-radix-views-block--excerpted-from-block-1" 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position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="National Poetry Month Themed Packet for High School" aria-label="input"> <div id="favoritesResponseError-2347896"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/2347896" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Printable" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-2347896"> <table data-drupal-selector="edit-collections" id="edit-collections--3" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--3" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--3" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--3" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--3" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--3" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--3" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--3" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-2347896 js-form-submit form-submit btn" data-nid="2347896" data-drupal-selector="edit-submit" type="submit" id="edit-submit--3" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-2347896" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-wqxhzcorxvijrjltxtkwtspxpu-wpcc07bjgtu2xvbi" type="hidden" name="form_build_id" value="form-wQxHzcorxVIJRJlTxtKWTspXpu_Wpcc07bJgTU2xvbI" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-3" type="hidden" name="form_id" value="favorites_addcollection _3" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/poetry-month/national-poetry-month-themed-packet-high-school" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-2347896" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <picture><source srcset="/sites/default/files/styles/scale400w/public/2022-03/animal-farm-teaching-unit-kit.jpg.webp" media="all and (min-width: 768px)" type="image/webp"><img src="https://www.teachervision.com/sites/default/files/styles/scale400w/public/2022-03/animal-farm-teaching-unit-kit.jpg?itok=BGR62tWP" alt="lesson plans for animal farm - kit for a complete unit on the novel" class="d-block img-fluid mx-auto"></picture> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">LESSON PLANS</p> <h3 class="content-title h6 mb-2">Animal Farm Teaching Unit Kit</h3> <p class="card-text"> Everything you need to teach a full or partial unit on Animal Farm Help your students dive into the modern relevance, m... </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-2409506"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-2409506 needs-validation form" data-drupal-selector="favorites-addcollection-4" data-node-alias="/reading-literature/animal-farm-teaching-unit-kit" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-4" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--4" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--4" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Animal Farm Teaching Unit Kit" aria-label="input"> <div id="favoritesResponseError-2409506"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/2409506" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Printable" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-2409506"> <table data-drupal-selector="edit-collections" id="edit-collections--4" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--4" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--4" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--4" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--4" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--4" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--4" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--4" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-2409506 js-form-submit form-submit btn" data-nid="2409506" data-drupal-selector="edit-submit" type="submit" id="edit-submit--4" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-2409506" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-ncfzqtcc-2redqp34ts-3c1osfcqsmsvcjixyogaq0" type="hidden" name="form_build_id" value="form--NcfZQTcc-2redQp34Ts-3c1OSfcQSMSvcjIXYoGAq0" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-4" type="hidden" name="form_id" value="favorites_addcollection _4" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/reading-literature/animal-farm-teaching-unit-kit" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-2409506" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> </div> </div> <div id="block-tv-radix-tvrelatedresources" class="block block--tv-radix-tvrelatedresources my-4"> <h2>Related Resources</h2> <div class="row row-cols-1 row-cols-sm-2 row-cols-lg-3 g-4"> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <div class="list-img-thumbnail bg-orange h-100"> <figure class="d-flex justify-content-center align-items-center h-100"> <span class="small-thumb-teaching-resource"></span> </figure> </div> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">TEACHING RESOURCE</p> <h3 class="content-title h6 mb-2">Problem Solving: Draw a Picture</h3> <p class="card-text"> Problem Solving: Draw a Picture What Is It? The draw a picture strategy is a problem-solving technique in which students... </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-5242"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5242 needs-validation form" data-drupal-selector="favorites-addcollection-2" data-node-alias="/mathematics/problem-solving-draw-picture" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-2" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--2" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--2" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Problem Solving: Draw a Picture" aria-label="input"> <div id="favoritesResponseError-5242"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/5242" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Articles" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-5242"> <table data-drupal-selector="edit-collections" id="edit-collections--2" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--2" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--2" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--2" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--2" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--2" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--2" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--2" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-5242 js-form-submit form-submit btn" data-nid="5242" data-drupal-selector="edit-submit" type="submit" id="edit-submit--2" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-5242" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-gruip04zp-ocpsaic9imspwpd78qbdvtwwtgdnafqq" type="hidden" name="form_build_id" value="form-gruIp04ZP--ocpsAic9IMSpwPd78qbDvTWwTGDNaFqQ" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-2" type="hidden" name="form_id" value="favorites_addcollection _2" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/mathematics/problem-solving-draw-picture" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-5242" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <div class="list-img-thumbnail bg-orange h-100"> <figure class="d-flex justify-content-center align-items-center h-100"> <span class="small-thumb-teaching-resource"></span> </figure> </div> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">TEACHING RESOURCE</p> <h3 class="content-title h6 mb-2">Problem Solving: Choose the Operation</h3> <p class="card-text"> Problem Solving: Choose the Operation What Is It? The process of "choosing the operation" involves deciding which mathem... </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-5240"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5240 needs-validation form" data-drupal-selector="favorites-addcollection-3" data-node-alias="/operations/problem-solving-choose-the-operation" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-3" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--3" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--3" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Problem Solving: Choose the Operation" aria-label="input"> <div id="favoritesResponseError-5240"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/5240" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Articles" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-5240"> <table data-drupal-selector="edit-collections" id="edit-collections--3" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--3" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--3" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--3" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--3" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--3" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--3" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--3" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-5240 js-form-submit form-submit btn" data-nid="5240" data-drupal-selector="edit-submit" type="submit" id="edit-submit--3" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-5240" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-euk6maqjnaok-hyc-hrspvyorbly2phwyn1iwmfdjpo" type="hidden" name="form_build_id" value="form-euk6MAqJnaok-HyC_hRSPVYoRblY2phWyN1IwmFdJpo" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-3" type="hidden" name="form_id" value="favorites_addcollection _3" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/operations/problem-solving-choose-the-operation" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-5240" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <div class="list-img-thumbnail bg-orange h-100"> <figure class="d-flex justify-content-center align-items-center h-100"> <span class="small-thumb-teaching-resource"></span> </figure> </div> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">TEACHING RESOURCE</p> <h3 class="content-title h6 mb-2">Problem Solving: Simplify the Problem</h3> <p class="card-text"> Problem Solving: Simplify the Problem What Is It? Simplifying a mathematics problem is a strategy that often is used alo... </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-5241"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5241 needs-validation form" data-drupal-selector="favorites-addcollection-4" data-node-alias="/mathematics/problem-solving-simplify-the-problem" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-4" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--4" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--4" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Problem Solving: Simplify the Problem" aria-label="input"> <div id="favoritesResponseError-5241"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/5241" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Articles" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-5241"> <table data-drupal-selector="edit-collections" id="edit-collections--4" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--4" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--4" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--4" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--4" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--4" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--4" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--4" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-5241 js-form-submit form-submit btn" data-nid="5241" data-drupal-selector="edit-submit" type="submit" id="edit-submit--4" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-5241" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-b1-griwadtzzamwxxtgg363gnvazvsifscaeivg0sde" type="hidden" name="form_build_id" value="form-b1-GRIWAdtZZAmwXxtgg363GnvaZvsiFSCAEivG0SDE" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-4" type="hidden" name="form_id" value="favorites_addcollection _4" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/mathematics/problem-solving-simplify-the-problem" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-5241" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <div class="list-img-thumbnail bg-orange h-100"> <figure class="d-flex justify-content-center align-items-center h-100"> <span class="small-thumb-teaching-resource"></span> </figure> </div> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">TEACHING RESOURCE</p> <h3 class="content-title h6 mb-2">Estimating Unknown Quantities</h3> <p class="card-text"> Estimating Unknown Quantities What Is It? Estimation is the process of guessing the approximate value of a number. </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-5236"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5236 needs-validation form" data-drupal-selector="favorites-addcollection-5" data-node-alias="/professional-development/estimating-unknown-quantities" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-5" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--5" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--5" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Estimating Unknown Quantities" aria-label="input"> <div id="favoritesResponseError-5236"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/5236" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Articles" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-5236"> <table data-drupal-selector="edit-collections" id="edit-collections--5" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--5" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--5" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--5" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--5" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--5" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--5" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--5" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-5236 js-form-submit form-submit btn" data-nid="5236" data-drupal-selector="edit-submit" type="submit" id="edit-submit--5" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-5236" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-uuymqyqfcxaflafbybe7-djzrwclybr-xjjzlpwpg1s" type="hidden" name="form_build_id" value="form-uUYmqYqFcxaFlAfbybE7_dJzrWCLYBR-XjjzLPwPG1s" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-5" type="hidden" name="form_id" value="favorites_addcollection _5" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/professional-development/estimating-unknown-quantities" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-5236" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <div class="list-img-thumbnail bg-orange h-100"> <figure class="d-flex justify-content-center align-items-center h-100"> <span class="small-thumb-teaching-resource"></span> </figure> </div> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-orange">TEACHING RESOURCE</p> <h3 class="content-title h6 mb-2">Geometry</h3> <p class="card-text"> Geometry What Is It? Geometry is the study of two- and three-dimensional figures. </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-5238"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-5238 needs-validation form" data-drupal-selector="favorites-addcollection-6" data-node-alias="/professional-development/geometry" action="/professional-development/using-manipulatives" method="post" id="favorites-addcollection-6" accept-charset="UTF-8"> <details data-drupal-selector="edit-add" id="edit-add--6" class="js-form-wrapper form-wrapper"><summary class="position-relative z-2" role="button" aria-controls="edit-add--6" aria-expanded="false" aria-pressed="false"> <i class="fa-regular fa-heart text-primary me-2"></i>ADD TO FAVORITES </summary></details> <div class="details-wrapper position-absolute start-0 bottom-0 bg-white w-100 rounded-bottom-4 z-3"> <div class="details-content h-100 p-3 overflow-auto"> <input style="width: 90%" class="favorites-add-textfield form-control" data-drupal-selector="edit-title" type="hidden" name="title" value="Geometry" aria-label="input"> <div id="favoritesResponseError-5238"></div> <input data-drupal-selector="edit-path" type="hidden" name="path" value="/node/5238" class="form-control" aria-label="input"> <input data-drupal-selector="edit-ntype" type="hidden" name="ntype" value="Articles" class="form-control" aria-label="input"> <input data-drupal-selector="edit-query" type="hidden" name="query" value="" class="form-control" aria-label="input"> <div id="collection-wrapper-5238"> <table data-drupal-selector="edit-collections" id="edit-collections--6" class="responsive-enabled table" data-striping="1"> <thead> <tr> <td></td> <th>Add to Folder</th> </tr> </thead> <tbody> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-138906 js-form-item-collections-138906 form-no-label form-check"> <input data-drupal-selector="edit-collections-138906" type="checkbox" id="edit-collections-138906--6" name="collections[138906]" value="138906" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>creative writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-141401 js-form-item-collections-141401 form-no-label form-check"> <input data-drupal-selector="edit-collections-141401" type="checkbox" id="edit-collections-141401--6" name="collections[141401]" value="141401" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>children's book</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-145241 js-form-item-collections-145241 form-no-label form-check"> <input data-drupal-selector="edit-collections-145241" type="checkbox" id="edit-collections-145241--6" name="collections[145241]" value="145241" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>activities</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-150256 js-form-item-collections-150256 form-no-label form-check"> <input data-drupal-selector="edit-collections-150256" type="checkbox" id="edit-collections-150256--6" name="collections[150256]" value="150256" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>classroom tools</td> </tr> <tr class="odd"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151741 js-form-item-collections-151741 form-no-label form-check"> <input data-drupal-selector="edit-collections-151741" type="checkbox" id="edit-collections-151741--6" name="collections[151741]" value="151741" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>language arts and writing</td> </tr> <tr class="even"> <td><div class="js-form-item form-item js-form-type-checkbox form-item-collections-151926 js-form-item-collections-151926 form-no-label form-check"> <input data-drupal-selector="edit-collections-151926" type="checkbox" id="edit-collections-151926--6" name="collections[151926]" value="151926" class="form-checkbox form-check-input" aria-label="checkbox"> </div> </td> <td>vocabulary</td> </tr> </tbody> </table> </div><a class="open-create-folder" href="javascript:void(0)">Create new folder</a><span class="add-collection fw-bold">CREATE NEW FOLDER</span><div class="js-form-item form-item js-form-type-textfield form-item-cname js-form-item-cname form-no-label form-group my-3"> <input class="favorites-collection-name form-textfield form-control" placeholder="Folder Name" data-drupal-selector="edit-cname" type="text" id="edit-cname--6" name="cname" value="" size="20" maxlength="255" aria-label="input"> </div> <input class="btn-primary mt-2 me-2 addtofavorites-submit-5238 js-form-submit form-submit btn" data-nid="5238" data-drupal-selector="edit-submit" type="submit" id="edit-submit--6" name="op" value="Update"> <input data-drupal-selector="edit-form-type" type="hidden" name="form_type" value="-5238" class="form-control" aria-label="input"> </div> </div> <input autocomplete="off" data-drupal-selector="form-f9wn33q7kh2olz6yl4-vz6g47sjzd84c05nmrw5spga" type="hidden" name="form_build_id" value="form-f9WN33q7Kh2OLz6YL4_vZ6g47SJZd84C05nMRw5SpgA" class="form-control" aria-label="input"> <input data-drupal-selector="edit-favorites-addcollection-6" type="hidden" name="form_id" value="favorites_addcollection _6" class="form-control" aria-label="input"> </form> </div> </div> </div> <a class="stretched-link" href="/professional-development/geometry" hreflang="en" aria-label="Read more"></a> </div> <div id="favoritesResponse-5238" class="favorites-response position-fixed bottom-0 end-0 z-3 p-4 mw-100"></div> </div> <div class="col"> <div class="resource-card card h-100 overflow-hidden shadow-sm"> <div class="card-img-container overflow-hidden sh-3"> <picture><source srcset="/sites/default/files/styles/scale400w/public/2019-02/iStock-911026900.jpg.webp" media="all and (min-width: 768px)" type="image/webp"><img src="https://www.teachervision.com/sites/default/files/styles/scale400w/public/2019-02/iStock-911026900.jpg?itok=uXmo9xM0" alt="Teacher teaching Math" class="d-block img-fluid mx-auto"></picture> </div> <div class="card-body position-relative d-flex flex-column bg-white"> <p class="content-type mb-2 text-success">REFERENCE</p> <h3 class="content-title h6 mb-2">Teaching Mathematics to Gifted Students in a Mixed-Ability Classroom</h3> <p class="card-text"> From: The ERIC Clearinghouse on Disabilities and Gifted Education ERIC EC Digest #E594Author: Dana T. </p> </div> <div class="mt-auto bg-white z-2 p-3"> <div id="favorites-section-104046"> <div class="favorites-section-inner"> <form class="collection-wrapper-unique add-favorites-form-104046 needs-validation form" 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