differentiate problem solving from drills

• Get IGI Global News

• All Products
• Book Chapters
• Journal Articles
• Video Lessons
• Teaching Cases
• Recommend to Librarian
• Recommend to Colleague
• Fair Use Policy

• Access on Platform

Export Reference

• e-Journal Collection
• e-Book Collection Select
• Social Sciences Knowledge Solutions e-Journal Collection
• Computer Science and IT Knowledge Solutions e-Journal Collection

The Differences between Problem-Based and Drill and Practice Games on Motivations to Learn

Problem-Based education has been put forward as the most fruitful approach when it comes to serious game design (Aldrich, 2009; Gee, 2005). In Problem-Based learning, students start with a problem. This problem is rather loosely defined as something ‘for which an individual lacks a ready response’ (Hallinger, 1992, p. 27). Problem-Based education distinguishes between well- and ill-defined problems. Ill-defined problems are those ‘in which one or several aspects of the situation is not well specified, the goals are unclear, and there is insufficient information to solve them’ (Ge & Land, p5 in Ertmer et al., 2008). Shaffer’s (Shaffer, Squire, Halverson, & Gee, 2005; Shaffer, 2008) suggestion for epistemic games, in which players adopt the perspective of a professional to confront complex problems in simulation-like game, aligns with Problem-Based learning approach.

Drill & Practice learning teaches the ‘what’ and the ‘when’, but not the ‘why’ and the ‘how’. Ke (2008) suggests that students in Drill & Practice Learning merely memorize facts. As a result, this kind of learning may not facilitate creative thought or stimulate problem-solving skills. Or, as Reeve et al. (2004) state, it may not present students with the opportunity to experiment, explore and struggle with the learning content to find the truth for themselves. Games such as Math Gran Prix (Atari Inc., 1982), Math Blaster (Davidson & Associates, 1994), and Dr. Kawashima’s Brain Training (Nintendo SDD, 2005) align with the Drill & Practice learning. In these games, there is only one solution to a mathematical challenge, and players are prompted to input the correct one.

Complete Article List

• Exponent Laws and Logarithm Laws
• Trig Formulas and Identities
• Differentiation Rules
• Trig Function Derivatives
• Table of Derivatives
• Table of Integrals
• Calculus Home

Calculate Derivatives – Problems & Solutions

Are you working to calculate derivatives in Calculus? Let’s solve some common problems step-by-step so you can learn to solve them routinely for yourself.

Jump down this page to: [ Power rule: $x^n$ ] [ Exponential: $e^x$ ] [ Trig derivs ] [ Product rule ] [ Quotient rule ] [ More problems & University exam problems ] [ Chain rule (will take you to a new page)]

Update: We now have a much more step-by-step approach to helping you learn how to compute even the most difficult derivatives routinely, inclduing making heavy use of interactive Desmos graphing calculators so you can really learn what’s going on. Please visit our Calculating Derivatives Chapter to really get this material down for yourself.

It’s all free, and waiting for you! (Why? Just because we’re educators who believe you deserve the chance to develop a better understanding of Calculus for yourself, and so we’re aiming to provide that. We hope you’ll take advantage!)

If you just need practice calculating derivative problems for now, previous students have found what’s below super-helpful. And if you have questions, please ask on our Forum ! It’s also free for your use.

Exponential

Trigonometric.

Notice that a negative sign appears in the derivatives of the co-functions: cosine, cosecant, and cotangent.

Constant Factor Rule

For example, $\dfrac{d}{dx}\left(4x^3\right) = 4 \dfrac{d}{dx}\left(x^3 \right) =\, … $

Sum of Functions Rule

For example, $\dfrac{d}{dx}\left(x^2 + \cos x \right) = \dfrac{d}{dx}\left( x^2\right) + \dfrac{d}{dx}(\cos x) = \, …$

Product Rule

Iv. quotient rule.

Many students remember the quotient rule by thinking of the numerator as “hi,” the demoninator as “lo,” the derivative as “d,” and then singing

“lo d-hi minus hi d-lo over lo-lo”

☕ Buy us a coffee We're working to add more, and would appreciate your help to keep going! 😊

Careful: the derivative of $fg$ is NOT simply the product of $\dfrac{df}{dx}$ and $\dfrac{dg}{dx}.$ Please review the Product Rule Examples, and then answer again.

Oops: recall that $\dfrac{d}{dx}\left( \sin x \right)=\cos x$. Please answer again.

The Product Rule states: $\dfrac{d}{dx}\left( fg \right)=\dfrac{df}{dx}\cdot g+f\cdot \dfrac{dg}{dx}$. Here, $f=x$, and $g=\sin x$. Please answer again.

"lo d-hi minus hi d-lo over lo-lo"

☕ Buy us a coffee If we've helped, please consider giving a little something back. Thank you! 😊

• What questions do you have about the solutions above?
• Which ones are giving you the most trouble?
• What other problems are you trying to work through for your class about calculating derivatives that you don't know how to do easily yet?

Share a link to this screen:

• Share on Facebook
• Share on Pinterest
• Email this Page
• Share on Reddit
• Share on Telegram
• Share on WhatsApp
• Share on SMS

• Terms of Use
• Privacy Policy

Matheno ®

Berkeley, California

AP ® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, this site.

© 2014–2024 Matheno, Inc.

• Your answers to multiple choice questions;
• Your self-chosen confidence rating for each problem, so you know which to return to before an exam (super useful!);
• Your progress, and specifically which topics you have marked as complete for yourself.

Your selections are for your use only, and we do not share your specific data with anyone else. We do use aggregated data to help us see, for instance, where many students are having difficulty, so we know where to focus our efforts.

You will also be able to post any Calculus questions that you have on our Forum , and we'll do our best to answer them!

We believe that free, high-quality educational materials should be available to everyone working to learn well. There is no cost to you for having an account, other than our gentle request that you contribute what you can, if possible, to help us maintain and grow this site.

Please join our community of learners. We'd love to help you learn as well as you can!

Differentiated Problem Solving: A New Approach

I know so many of you have been looking for a way to build deep math thinking with your intermediate students–I know this because I get questions about it all the time!  

 You want your students to be challenged in new and interesting ways—and be easily able to differentiate so that ALL your students can benefit, right? Here’s the problem…what’s challenging for some is way too tricky for others, right?  Or you want to give a task–and you do–and then 1/3 of your class is finished in minutes and asking for more while others have barely gotten started.  (Tell me that I’m not the only one this happens to!)

I often get people asking me what a “typical” math class looks like in my room–and I have to be honest.  THere is no such thing.  I feel like I operate on a menu system…I have all these “tasty” things and I serve them up when I think it makes the most sense!  That being said, here are a few suggestions for working these quality tasks into your day. *Use as a “bell ringer” or warm up task. The goal of this should not be getting a correct answer, but the actual WORK of doing the problem solving! Each question has a starting point which can be used whole-class (you choose how much modeling/help) and then parts 2 and 3 can be used for everyone—or just for students who are ready! The colored slides are perfect to project from your computer…you can click to the next slides to show parts 2 and 3…but if students aren’t ready, no big deal! The original problem appears on every slide! *Print and laminate and use as task cards at a problem solving station. These half-page cards are low ink and are perfect for math rotations, math workshop, guided math, or for fast finishers.  Whether you do rotations or organize your stations differently, having quality problems ready to go really saves your time. *Use as a reproducible problem-solving journal. I typed these problems up in a journal format with a full page of work space for the first part of the problem–with parts 2 and 3 copied on the next page. Copying the entire journal only takes 12 pieces of paper (without the cover) and is full of the 36 tasks. These can be used in so many ways—and even flexibly within a given classroom. I have some students who only do part of the collections–and others who might have the time and motivation to do much, much more. *Consider using open-ended tasks as enrichment opportunities for students needing just a bit more. This is perfect–whether you have one student or a handful.  They can work together, practice that accountable math talk, and push each other.

Problem solving is not easy!

Like many of my resources, this set of problems is certainly not meant to be a time filler! It is meant to be a rich and meaningful problem solving experience for you to use with your students. HOW you use it is up to you!  I know we are all busy…but the time we invest in modeling some of the thinking and strategies needed with this type of problem REALLY pays off in the long run as students become more and more independent. 

When I use tasks like this with MY students, there are a few things I like to make very clear and I think really contribute to building a culture for problem solving.  One of the most important things that I think teachers need to keep in mind is that we often “overteach”.  We TELL too much.  We push our own strategies and ideas onto them–even if they aren’t quite ready for them.  Before I used this problem with my students, I thought about how I would solve this as an adult–and then thought about what might get students off track.  In this case…I did NOT want to show students my “boxes” (although for a few students I did coach them in this direction after they had worked awhile), but I DID want to make sure students understood the task–and the terms (like digits) so that they didn’t waste time.  I didn’t TELL them what the task was…but students worked in pairs to make sense of it, then we came back as a whole group to discuss it and get on the same page.

Push Yourself!

Interested in checking out this set of problems?  Just click HERE or the image below.

Want to pin this for later?  Here you go!

SHARE THIS POST:

The Power of Morning Meetings in Elementary School

Building Excellence: Helping Elementary Students Understand Quality Work

How Project Based Learning Can Help Students Love Math

Quick Links

• The Teacher Studio 2024
• Site Design By Laine Sutherland Designs

Questions about FAFSA and CADAA?

Visit our Financial Aid and Scholarship Office for updated information, workshops and FAQs.

Earthquake Advisory

Campus closed until further notice

Mourtos, Nikos J

Site navigation, problem-solving vs. exercise solving.

• Give to SJSU

Problem-solving

• Involves a process used to obtain a best answer to an unknown, subject to some constraints.
• The situation is ill defined. There is no problem statement and there is some ambiguity in the information given. Students must define the problem themselves. Assumptions must be made regarding what is known and what needs to be found.
• The context of the problem is brand new (i.e., the student has never encountered this situation before).
• There is no explicit statement in the problem that tells the student what knowledge / technique / skill to use in order to solve the problem.
• There may be more than one valid approach.
• The algorithm for solving the problem is unclear.
• Integration of knowledge from a variety of subjects may be necessary to address all aspects of the problem.
• Requires strong oral / written communication skills to convey the essence of the problem and present the results.

Exercise Solving

• Involves a process to obtain the one and only right answer for the data given.
• The situation is well defined. There is an explicit problem statement with all the necessary information (known and unknown).
• The student has encountered similar exercises in books, in class or in homework.
• Exercises often prescribe assumptions to be made, principles to be used and sometimes they even give hints.
• There is usually one approach that gives the right answer.
• A usual method is to recall familiar solutions from previously solved exercises.
• Exercises involve one subject and in many cases only one topic from this subject.
• Communication skills are not essential, as most of the solution involves math and sketches.

SJSU Links and Resources

Information for.

• Current Students
• Faculty and Staff
• Future Students
• Researchers
• Engineering
• Graduate Studies
• Health and Human Sciences
• Humanities and the Arts
• Professional Education
• Social Sciences

Quick Links

• Budget Central
• Careers and Jobs
• Emergency Food & Housing
• Faculty & Staff
• Freedom of Speech
• King Library
• Land Acknowledgement
• Parenting Students
• Parking and Maps
• Annual Security Report [pdf]
• Contact Form
• Doing Business with SJSU
• File a Complaint
• Report a Title IX Complaint

San José State University One Washington Square, San José, CA 95192

408-924-1000

• PROFESSIONAL HISTORY
• BOOK REVIEWS
• CONSULTANCY REPORTS
• CORE FRAMES

The Three point problem: Calculating strike and dip from multiple DD Holes

The Three Point Problem

In a previous post (see  here ) I described how quantitative orientation data can be collected from from a single drill hole, even where the core is not oriented.  In this post, techniques for collecting orientation data on planes are described when more than one non-oriented hole is available from a prospect.

The need to determine the strike and dip of a planar structure from a number of drill intersections is one that occurs very frequently – this is often called the three-point problem [1] and every geologist should be familiar with the simple solutions to it (Marjoribanks, 2007).

The attitude of any plane is fully defined if the position in 3D space of three or more points on that surface is known. Where three separate holes intersect the same marker bed, they provide three points of known position on that surface. From the intercept data, there are three ways of calculating the strike and dip. The first makes use of structure contours. The second involves the use of a stereonet. In the presentation of the solutions below, it is assumed that the bed to be measured has the same attitude in all three holes.

  There is, of course, the third way: feed the numbers into a computer program such as the freeware you can download at  www.edumine.com  ; then click the button marked “Answer”.  The “answer” in this case is useful strike/dip/dip direction information from intersection coordinates.  But what matters almost as much as information is the means by which you arrive at it.  Graphical manipulation is a mental process that engages your brain in the 3D realities of your measurements, and it is this pre-conditioning that helps you turn mere information into knowledge and understanding.  In this case, “Understanding” is the ability to create predictive models for ore, or to cut losses, walk away, and try your luck elsewhere.  For this kind of understanding, easy shortcuts can be self-defeating.   

Remember the mantra, the theme of many of my posts:

Data is not information

Information is not knowledge

Knowledge is not understanding

Understanding is not wisdom

  Solution using structure contours.

  In figure 1, the intersections of three holes into a common marker bed are projected onto a plan. Proceed as follows:

  Figure 1 Using structure contours to determine the strike and dip of a planar surface intersected in three drill holes. On the map three drill holes have intercepted a common bed. The intercepts have been projected vertically onto the plan and labelled with their height above a common datum. Click for full size figure.

Determine the three-dimensional coordinates (i.e. northing, easting and height above the datum) of each intersection of the marker bed in the holes.

Plot the three bed intersection points on a map using the northing and easting coordinates for each intersection. Write the depth (often called the Relative Level, or RL) of the intersection beside each point.

On the map draw a lines joining the hole intersections (figure 2). The height of the intersection at the beginning and the end of each line is already marked. Using a ruler, scale off along each line to identify the positions of all intermediate depths: identify and mark even-number depth divisions.  

Draw the lines joining points of equal depth on the surface. The lines correspond to the height contours on a topographic map and are known as structure contours. They represent the plot of horizontal lines on the bed and thus mark its strike.  This strike can then be measured on the map using a protractor. 

Use the map scale to measure the horizontal distance (h) between any two contour lines – the further apart the better. Since the vertical separation (v) of the contour lines is known, the dip of the surface (d) can be calculated according to the formula:

  Tan d = v/h

Figure 2 Constructing lines between each plotted intersection, the position of different heights along the lines can be scaled off. Structure contour lines (dashed) for the bed are constructed by joining points of equal height. These lines define the strike of the surface. From the map scale, the horizontal distance between lines of known height can be measured – simple trigonometry then allows the dip to be calculated. Click for full size figure.

  Solution using a stereonet

  Proceed as follows (see figures 3 and 4):

Determine the absolute position coordinates (i.e. northing, easting and height above a common datum) of each intersection of the marker bed in the three holes.

Figure .3 Using a stereonet to determine the strike and dip of planar surface intersected in three drill holes. The intercepts are projected onto a plan and labelled with the height of each intersection above a common datum. The line that joins any pair of intersections is an apparent dip on the bed and can be described by its dip and dip direction. Where v is the vertical height difference between each pair of intersections, and h their horizontal separation, the apparent dip ( a ) can be calculated using Tan a = v/h. The dip direction is measured directly from the plan with a protractor. Three apparent dips can be calculated in this way. Click for full size figure.

Plot the three intersection points on a map. Use a protractor to measure the trend (bearing) of the lines joining the three points. Use a ruler to scale off the horizontal distance between the points. Knowing the horizontal and elevation difference between any pairs of intersection points, simple trigonometric formulae (see step 5 above) will provide the angle of plunge (the angle which the line makes with the horizontal, measured in the vertical plane) for the line that joins any two pairs of points.

We have now calculated the trend and plunge of three lines lying on the surface of the marker bed. Mark these lines on to a stereonet overlay. They plot as three points, as shown on figure 3.

Rotate the overlay so as to bring the three points to lie on a common great circle. Only one great circle will satisfy all three points [2] . This great circle represents the trace of the bed that was intersected by the drill holes.

From the net, read off the strike and dip of the surface (or dip and dip direction, or apparent dip on any given drill section).    

  An elegant stereonet solution to determining the attitude of planes in non-oriented core

Where there is no single marker bed that can be correlated between adjacent holes, it is sometimes still possible to determine the orientation of a set of parallel surfaces (such as bedding planes, a cleavage, or a vein set) provided that the surfaces have been cored by a minimum of three nonparallel drill holes (Mead, 1921, Bucher, 1943). The same technique can even be extended to a single hole, provided that the hole has sufficient deviation along its length for the differently oriented sectors of the same hole to be considered in the same way as three separate holes (Laing, 1977).

In the example illustrated in the following figures, three adjacent but non-parallel angle holes have intersected the same set of parallel, planar quartz veinlets. None of the core is oriented, but the average alpha (α) angle (for definition of alpha angle see here ) between the veins and the core axis has been measured in each hole: it is 10° in Hole 1; 56° in Hole 2 and 50° in Hole 3.

In our example, Hole 1 is drilled at - 50° to 270° ; Hole 2 at - 65° to 090° and Hole 3 at - 60° to 345 °. On the stereonet, the orientation of each drill hole plots as a point.

When plotting planes on a stereonet it is always much easier to work with the pole [3] to the plane rather than the plane itself. If a plane makes an angle α with the core axis, then the pole to the plane makes an angle of 90 – α to the core axis, as illustrated in figure 5.

Figure.5  The angular relationship of the alpha angle (α) to the pole (i.e. the normal) of a surface intersected in drill core. Click for full size figure.

  Let us consider Hole 1 (figure 6).  Angle α for the vein set is known. Because the core is not oriented, the poles to the veins could lie anywhere within the range of orientations that is produced as the core is rotated one complete circle about its long axis.  This range defines a cone, centered on the core axis, with an apical angle of 2 x α. From figure 5, we see  that the pole to the plane will describe a cone with an apical angle of 2 x (90 – α). That is all we can tell from one hole, but this information can be shown on the stereonet, because a cone centered on a drill hole plots as a small circle girdle around that hole. In Hole 1, 90 – α is 80°. The vein set in Hole 1 can therefore be represented by a small circle girdle at an angle of 80° to the hole plot. 

Now the same procedure is carried out for Hole 2 by drawing a small circle at 90-α (34°) to the plot of that hole on the net (figure 7). The small circle about Hole 1 and the small circle about Hole 2 intersect at two points (P1 & P2) – these points represent two possible orientations for the vein set. Already, with just two DD holes we have reduced the problem of determining the orientation of the vein set from an infinite number of possibilities to just two possibilities.

Now, in the same manner, we draw the third small circle about Hole 3 representing the alpha angle measured in that hole (figure 8). We now have three small circle girdles on our net, centered about each of the three drill holes. Since the assumption behind this procedure is that all measurements are of the one vein set with a constant orientation, the single point (P) where the three small girdles intersect must represent the pole to the one orientation that is common to all three holes. This pole defines the attitude of the common vein set seen in the holes. Of course, with a real set of measurements it is unlikely that three small circles plotted in this way would meet at a single point. Rather, the intersecting lines will define a triangle whose size reflects the accuracy of the measurements (and the assumptions made that we are dealing with a single parallel set of surfaces). The true pole position (if there is one) will lie somewhere within this triangle of error.

  From the point P, the strike and dip (or dip and dip direction, or apparent dip on drill section) of the vein set can be simply read off from the net.

Figure 8   On the stereonet, the poles to all possible planes making an angle of α  degrees to the axis of  Hole 3  inscribe a small circle girdle at 90-α (40°) to the plot of the hole.  The small circles about Hole 1, Hole2 & Hole 3 intersect at a single P. P is the pole to the unique plane which satisfies the measurements made in the three holes.. Click for larger image.

  Bucher WH (1943) Dip and strike for three not parallel drill holes lacking key beds. Econ Geol 38, 648-657.  

Laing WP (1977) Structural interpretation of drill core from folded and cleaved rocks. Econ Geol 72, 671-685.  

Marjoribanks RW (2007) Structural logging of drill core. Australian Institute of Geologists Handbook 5 (2 nd ed.), 68p.

  Mead WJ (1921) Determination of the attitude of concealed bedding formations by diamond drilling. Econ Geol 21, 37-47.

[1] Presumably in a conscious or subconscious nod to Sherlock Holmes’ “3-pipe problem”. In Holmes case, it was an amount of opium, but there is no need to resort to such extreme measures to solve the 3-point problem.

[2] Actually, because a plotted point on a stereonet represents the orientation of a line, only two such points are needed to define the plane on which they lie.  The use of a third point (line) adds accuracy and provides for error checking.

[3] The pole to a plane is the line at right angles, or normal, to the plane. By plotting the pole, the attitude of a plane can be represented on a stereonet by a single point.

Note that this blog no longer accepts on-line comments (there was too much spam coming in!). However I welcome all questions, comments or criticisms. You can send these to me via the CONTACT ME  tab. 

Comments are closed.

• Diamond Drilling
• Epistemology
• Geochemistry
• Geological Mapping
• Geology Profession
• Historyof Geology
• Philosophy of Mineral Exploration
• Sterenet Solutions in structural geology
• Structural Geology
• Uncategorized

• CORE CURRICULUM 
• LITERACY > CORE CURRICULUM  > Into Literature, 6-12" data-element-type="header nav submenu" title="Into Literature, 6-12" aria-label="Into Literature, 6-12"> Into Literature, 6-12
• LITERACY > CORE CURRICULUM  > Into Reading, K-6" data-element-type="header nav submenu" title="Into Reading, K-6" aria-label="Into Reading, K-6"> Into Reading, K-6
• INTERVENTION
• LITERACY > INTERVENTION > English 3D, 4-12" data-element-type="header nav submenu" title="English 3D, 4-12" aria-label="English 3D, 4-12"> English 3D, 4-12
• LITERACY > INTERVENTION > Read 180, 3-12" data-element-type="header nav submenu" title="Read 180, 3-12" aria-label="Read 180, 3-12"> Read 180, 3-12
• SUPPLEMENTAL 
• LITERACY > SUPPLEMENTAL  > A Chance in the World SEL, 8-12" data-element-type="header nav submenu" title="A Chance in the World SEL, 8-12" aria-label="A Chance in the World SEL, 8-12"> A Chance in the World SEL, 8-12
• LITERACY > SUPPLEMENTAL  > Amira Learning, K-6" data-element-type="header nav submenu" title="Amira Learning, K-6" aria-label="Amira Learning, K-6"> Amira Learning, K-6
• LITERACY > SUPPLEMENTAL  > Classcraft, K-8" data-element-type="header nav submenu" title="Classcraft, K-8" aria-label="Classcraft, K-8"> Classcraft, K-8
• LITERACY > SUPPLEMENTAL  > JillE Literacy, K-3" data-element-type="header nav submenu" title="JillE Literacy, K-3" aria-label="JillE Literacy, K-3"> JillE Literacy, K-3
• LITERACY > SUPPLEMENTAL  > Waggle, K-8" data-element-type="header nav submenu" title="Waggle, K-8" aria-label="Waggle, K-8"> Waggle, K-8
• LITERACY > SUPPLEMENTAL  > Writable, 3-12" data-element-type="header nav submenu" title="Writable, 3-12" aria-label="Writable, 3-12"> Writable, 3-12
• LITERACY > SUPPLEMENTAL  > ASSESSMENT" data-element-type="header nav submenu" title="ASSESSMENT" aria-label="ASSESSMENT"> ASSESSMENT
• CORE CURRICULUM
• MATH > CORE CURRICULUM > Arriba las Matematicas, K-8" data-element-type="header nav submenu" title="Arriba las Matematicas, K-8" aria-label="Arriba las Matematicas, K-8"> Arriba las Matematicas, K-8
• MATH > CORE CURRICULUM > Go Math!, K-6" data-element-type="header nav submenu" title="Go Math!, K-6" aria-label="Go Math!, K-6"> Go Math!, K-6
• MATH > CORE CURRICULUM > Into Algebra 1, Geometry, Algebra 2, 8-12" data-element-type="header nav submenu" title="Into Algebra 1, Geometry, Algebra 2, 8-12" aria-label="Into Algebra 1, Geometry, Algebra 2, 8-12"> Into Algebra 1, Geometry, Algebra 2, 8-12
• MATH > CORE CURRICULUM > Into Math, K-8" data-element-type="header nav submenu" title="Into Math, K-8" aria-label="Into Math, K-8"> Into Math, K-8
• MATH > CORE CURRICULUM > Math Expressions, PreK-6" data-element-type="header nav submenu" title="Math Expressions, PreK-6" aria-label="Math Expressions, PreK-6"> Math Expressions, PreK-6
• MATH > CORE CURRICULUM > Math in Focus, K-8" data-element-type="header nav submenu" title="Math in Focus, K-8" aria-label="Math in Focus, K-8"> Math in Focus, K-8
• SUPPLEMENTAL
• MATH > SUPPLEMENTAL > Classcraft, K-8" data-element-type="header nav submenu" title="Classcraft, K-8" aria-label="Classcraft, K-8"> Classcraft, K-8
• MATH > SUPPLEMENTAL > Waggle, K-8" data-element-type="header nav submenu" title="Waggle, K-8" aria-label="Waggle, K-8"> Waggle, K-8
• MATH > INTERVENTION > Math 180, 3-12" data-element-type="header nav submenu" title="Math 180, 3-12" aria-label="Math 180, 3-12"> Math 180, 3-12
• SCIENCE > CORE CURRICULUM  > Into Science, K-5" data-element-type="header nav submenu" title="Into Science, K-5" aria-label="Into Science, K-5"> Into Science, K-5
• SCIENCE > CORE CURRICULUM  > Into Science, 6-8" data-element-type="header nav submenu" title="Into Science, 6-8" aria-label="Into Science, 6-8"> Into Science, 6-8
• SCIENCE > CORE CURRICULUM  > Science Dimensions, K-12" data-element-type="header nav submenu" title="Science Dimensions, K-12" aria-label="Science Dimensions, K-12"> Science Dimensions, K-12
• SCIENCE > READERS > ScienceSaurus, K-8" data-element-type="header nav submenu" title="ScienceSaurus, K-8" aria-label="ScienceSaurus, K-8"> ScienceSaurus, K-8
• SOCIAL STUDIES > CORE CURRICULUM  > HMH Social Studies, 6-12" data-element-type="header nav submenu" title="HMH Social Studies, 6-12" aria-label="HMH Social Studies, 6-12"> HMH Social Studies, 6-12
• SOCIAL STUDIES > SUPPLEMENTAL > Writable" data-element-type="header nav submenu" title="Writable" aria-label="Writable"> Writable
• For Teachers 
• PROFESSIONAL DEVELOPMENT > For Teachers  > Coachly" data-element-type="header nav submenu" title="Coachly" aria-label="Coachly"> Coachly
• PROFESSIONAL DEVELOPMENT > For Teachers  > Teacher's Corner" data-element-type="header nav submenu" title="Teacher's Corner" aria-label="Teacher's Corner"> Teacher's Corner
• PROFESSIONAL DEVELOPMENT > For Teachers  > Live Online Courses" data-element-type="header nav submenu" title="Live Online Courses" aria-label="Live Online Courses"> Live Online Courses
• PROFESSIONAL DEVELOPMENT > For Teachers  > Program-Aligned Courses" data-element-type="header nav submenu" title="Program-Aligned Courses" aria-label="Program-Aligned Courses"> Program-Aligned Courses
• For Leaders
• PROFESSIONAL DEVELOPMENT > For Leaders > The Center for Model Schools (formerly ICLE)" data-element-type="header nav submenu" title="The Center for Model Schools (formerly ICLE)" aria-label="The Center for Model Schools (formerly ICLE)"> The Center for Model Schools (formerly ICLE)
• MORE > undefined > Assessment" data-element-type="header nav submenu" title="Assessment" aria-label="Assessment"> Assessment
• MORE > undefined > Early Learning" data-element-type="header nav submenu" title="Early Learning" aria-label="Early Learning"> Early Learning
• MORE > undefined > English Language Development" data-element-type="header nav submenu" title="English Language Development" aria-label="English Language Development"> English Language Development
• MORE > undefined > Homeschool" data-element-type="header nav submenu" title="Homeschool" aria-label="Homeschool"> Homeschool
• MORE > undefined > Intervention" data-element-type="header nav submenu" title="Intervention" aria-label="Intervention"> Intervention
• MORE > undefined > Literacy" data-element-type="header nav submenu" title="Literacy" aria-label="Literacy"> Literacy
• MORE > undefined > Mathematics" data-element-type="header nav submenu" title="Mathematics" aria-label="Mathematics"> Mathematics
• MORE > undefined > Professional Development" data-element-type="header nav submenu" title="Professional Development" aria-label="Professional Development"> Professional Development
• MORE > undefined > Science" data-element-type="header nav submenu" title="Science" aria-label="Science"> Science
• MORE > undefined > undefined" data-element-type="header nav submenu">
• MORE > undefined > Social and Emotional Learning" data-element-type="header nav submenu" title="Social and Emotional Learning" aria-label="Social and Emotional Learning"> Social and Emotional Learning
• MORE > undefined > Social Studies" data-element-type="header nav submenu" title="Social Studies" aria-label="Social Studies"> Social Studies
• MORE > undefined > Special Education" data-element-type="header nav submenu" title="Special Education" aria-label="Special Education"> Special Education
• MORE > undefined > Summer School" data-element-type="header nav submenu" title="Summer School" aria-label="Summer School"> Summer School
• BROWSE RESOURCES
• BROWSE RESOURCES > Classroom Activities" data-element-type="header nav submenu" title="Classroom Activities" aria-label="Classroom Activities"> Classroom Activities
• BROWSE RESOURCES > Customer Success Stories" data-element-type="header nav submenu" title="Customer Success Stories" aria-label="Customer Success Stories"> Customer Success Stories
• BROWSE RESOURCES > Digital Samples" data-element-type="header nav submenu" title="Digital Samples" aria-label="Digital Samples"> Digital Samples
• BROWSE RESOURCES > Events" data-element-type="header nav submenu" title="Events" aria-label="Events"> Events
• BROWSE RESOURCES > Grants & Funding" data-element-type="header nav submenu" title="Grants & Funding" aria-label="Grants & Funding"> Grants & Funding
• BROWSE RESOURCES > International" data-element-type="header nav submenu" title="International" aria-label="International"> International
• BROWSE RESOURCES > Research Library" data-element-type="header nav submenu" title="Research Library" aria-label="Research Library"> Research Library
• BROWSE RESOURCES > Shaped - HMH Blog" data-element-type="header nav submenu" title="Shaped - HMH Blog" aria-label="Shaped - HMH Blog"> Shaped - HMH Blog
• BROWSE RESOURCES > Webinars" data-element-type="header nav submenu" title="Webinars" aria-label="Webinars"> Webinars
• CUSTOMER SUPPORT
• CUSTOMER SUPPORT > Contact Sales" data-element-type="header nav submenu" title="Contact Sales" aria-label="Contact Sales"> Contact Sales
• CUSTOMER SUPPORT > Customer Service & Technical Support Portal" data-element-type="header nav submenu" title="Customer Service & Technical Support Portal" aria-label="Customer Service & Technical Support Portal"> Customer Service & Technical Support Portal
• CUSTOMER SUPPORT > Platform Login" data-element-type="header nav submenu" title="Platform Login" aria-label="Platform Login"> Platform Login
• Learn about us
• Learn about us > About" data-element-type="header nav submenu" title="About" aria-label="About"> About
• Learn about us > Diversity, Equity, and Inclusion" data-element-type="header nav submenu" title="Diversity, Equity, and Inclusion" aria-label="Diversity, Equity, and Inclusion"> Diversity, Equity, and Inclusion
• Learn about us > Environmental, Social, and Governance" data-element-type="header nav submenu" title="Environmental, Social, and Governance" aria-label="Environmental, Social, and Governance"> Environmental, Social, and Governance
• Learn about us > News Announcements" data-element-type="header nav submenu" title="News Announcements" aria-label="News Announcements"> News Announcements
• Learn about us > Our Legacy" data-element-type="header nav submenu" title="Our Legacy" aria-label="Our Legacy"> Our Legacy
• Learn about us > Social Responsibility" data-element-type="header nav submenu" title="Social Responsibility" aria-label="Social Responsibility"> Social Responsibility
• Learn about us > Supplier Diversity" data-element-type="header nav submenu" title="Supplier Diversity" aria-label="Supplier Diversity"> Supplier Diversity
• Join Us > Careers" data-element-type="header nav submenu" title="Careers" aria-label="Careers"> Careers
• Join Us > Educator Input Panel" data-element-type="header nav submenu" title="Educator Input Panel" aria-label="Educator Input Panel"> Educator Input Panel
• Join Us > Suppliers and Vendors" data-element-type="header nav submenu" title="Suppliers and Vendors" aria-label="Suppliers and Vendors"> Suppliers and Vendors
• Divisions > Center for Model Schools (formerly ICLE)" data-element-type="header nav submenu" title="Center for Model Schools (formerly ICLE)" aria-label="Center for Model Schools (formerly ICLE)"> Center for Model Schools (formerly ICLE)
• Divisions > Heinemann" data-element-type="header nav submenu" title="Heinemann" aria-label="Heinemann"> Heinemann
• Divisions > NWEA" data-element-type="header nav submenu" title="NWEA" aria-label="NWEA"> NWEA
• Platform Login

HMH Support is here to help you get back to school right. Get started

SOCIAL STUDIES

PROFESSIONAL DEVELOPMENT

Differentiated Instruction

7 Strategies for Differentiated Math Instruction

Math classrooms are mosaics of strengths and experiences. When we have students with diverse backgrounds—with various languages, achievements, and interests—in the same space, everyone learns from each other and broadens their world.

On the flip side, though, teaching math to a broad array of students can be challenging. Do you struggle to reach all of your students? Are you a newer teacher who is looking to improve your practice? The strategies for differentiated instruction provided here might help you out.

What Is Differentiated Math Instruction?

Differentiated math instruction refers to the collection of techniques, strategies, and adaptations you can use to reach your diverse group of learners and make mathematics accessible to every single one. Dr. Timothy Kanold , former president of the National Council of Supervisors of Mathematics (NCSM)—and HMH author— clarifies that differentiation in a math lesson is “differentiation on the entry points into the task for support or the exit point to advance student thinking.”

By applying various tools and strategies, such as incorporating technology, assigning hands-on projects, and teaching in math small-group formats, you can help every student meet expectations. We know that there are different schools of thought regarding what differentiation means. When we use the term, we are talking about providing student choice, voice, and agency. Differentiating instruction isn’t meant to add more work to your day. Quite the opposite, in fact; it’s meant as a teaching approach that will help you to reach more students in terms of accessibility and equity, making your job both easier and more effective in the long run.

Why Is Differentiating Math Instruction Important?

Some people think that math, more than any other subject, is the best fit for differentiation. Even though a 2018 survey by Texas Instruments found that 46% of kids said they really liked math, there are hundreds of books, websites, and memes discussing the difficulty of the subject. From the anxiety caused by there being only one correct answer to the cultural buy-in to the myth of being—or not being—a “math person” to the fear of solving a word problem, many students struggle with math. In addition, many students and educators alike find it hard to make the connection between math and the real world, which only increases disillusionment with the subject. That’s why it’s especially important to be open to new ways of providing instruction .

The National Council of Teachers of Mathematics (NCTM) promotes differentiating math instruction for differences in learning as well as differences in achievement, interest, and confidence. NCTM advises that the need is greater in middle and high school, as higher-level math relies on more complex reasoning. When you differentiate your math instruction, you support all learners by targeting and addressing specific needs of groups and individual students.

Examples of Differentiated Instruction in Math

Do you need ideas for how to differentiate your teaching to be sure your math students are progressing? Below are seven differentiation strategies for math instruction, along with ways that you can use them in your math classroom. They serve as examples of differentiated instruction in math and may work better for some classrooms and math topics than others. Customize these ideas however you need to serve you and your students.

Strategy 1: Math Centers

For this, you’ll need to come up with a few activities your students can rotate through (be sure to browse our library of free activities and resources !), such as watching a video, reading an article, or solving a word problem. We spoke with Kristy McFarlane, an instructional supervisor at Sandshore Elementary School in New Jersey, about differentiation. She says math teachers at her school spend about 10 minutes on a mini-lesson for the whole class and then students spend about 15 minutes at various math centers. “They might meet with the teacher in a small group for extra help, use math software, do a game or project at the hands-on station, or do seat work based on the day’s mini-lesson,” she says.

Math centers are a powerful way to facilitate independent and small group learning within your classroom. Our Go Math! program, for example, is known for embedding resources and instructional time to math centers. If a select group of your students are all struggling to, say, add fractions, they may benefit from an activity that has them practice finding least common denominators. Think about ways to customize the groupings and centers so they’re perfect for your students’ strengths, misconceptions, and interests, and make use of tools that strategically group students and recommend activities for you.

Strategy 2: Activity Cards

Choice is an important part of differentiation, and letting students decide how they want to spend their time is a great way to appeal to various learning preferences. You’ll need to come up with math problems, tasks, or questions. As much as possible, use or create cards that span several lessons and offer options to work independently, with a partner, or in a small group. Ask for feedback so you can adjust future learning accordingly. Many of HMH’s math programs , including Into Math , Go Math! , and Into AGA include inquiry-based task and project cards that help teachers differentiate.

Strategy 3: Choice Boards

As we just mentioned, giving students the ability to make decisions about their learning is an important part of differentiation. A choice board is a graphic organizer that gives students activities to choose from. There are different types of choice boards, but they need to focus on specific learning needs, interests, and skills. Choice boards increase student ownership; students pace themselves and get to decide how to engage with information, along with how to demonstrate their learning. Some teachers create different versions of the same choice board; others will color-code options to signify topic, activity type, or expected level of challenge. Check out the choice board we developed for remote learning. This board covers all subjects but also includes a free template to get you started on a math-only version.

Strategy 4: Math Journals

Having students write about math is a great way for them to reflect on what they’ve learned and incorporate ELA instruction into the math classroom. Encourage your kids to summarize key points, answer open-ended questions, tie math into everyday experiences, or write about the most interesting or challenging math lesson. It’s also a way to provide an entry point for all students, including multilingual learners , as they can write a little or a lot in English or in their native language. Those who need extra support might be given sentence starters. Students might also be given the choice to illustrate their ideas instead of writing them. Similar to activity cards, math journals are included in many of HMH’s math programs, including Into Math and Into AGA .

Strategy 5: Learning Contracts

If metacognition is the ability to think about thinking—including about how you learn—we owe it to students to help them develop and expand their metacognitive skills. One way to do this is to work on learning contracts. Throughout the year, ask students to reflect on important lessons and set learning goals, including skills to learn or improve as well as new areas to explore. Use these learning contracts to help students learn to organize their thoughts. “One of our district’s goals is to have personalized learning opportunities for all students,” says McFarlane. “Each student creates a personalized success plan at the beginning of the year and does regular check-ins.” More broadly, metacognition is an idea that can be taught and practiced in the classroom and applies broadly to any subject.

Strategy 6: Math Games

Games are fun, motivational, and can help students deepen their mathematical reasoning. Some games encourage students to develop strategic and problem-solving skills or improve computational fluency. Seek out games where the math learning objective matches the game objective as a way for students to find joy in learning. Go Math! was designed to include both ready-made games for math centers and recommended games for differentiation within the teacher’s edition.

You can also use non-math games to provide a short mental break or a context for having math discussions. Look for ways to turn the game into mathematical discourse. How could you have scored more points? How much time did it take? What strategies did you use?

Strategy 7: Digital Math Practice

There are also lots of math apps and online tools that are designed to reinforce foundational understanding by allowing students to practice arithmetic and other math standards. In particular, seek out apps that are not simply timed drills with fun graphics, which are likely to make math anxiety worse for students who are not yet fluent in math facts. For digital math practice that extends far beyond just practicing arithmetic, our newest Go Math! for Grades K–6 has adaptive and personalized practice that aligns to our supplemental practice program, Waggle .

to you, consider giving your students problem-solving tasks with open-ended solutions. A single math problem can reveal different ways that students think about mathematics, which might be a less time-consuming way to assess student progress and determine an effective way to differentiate.

When you think critically about how to transform math instruction into differentiated math instruction, students will be more engaged because the content will be more relevant. They will achieve more success because they’ll be experiencing different types of activities, using various modalities, and contributing to the best of their abilities as they continue to grow.

HMH offers a variety of math classroom solutions to help you reach every student. Just looking for more articles and resources to help you differentiate math instructions? Try one of these to keep reading!

This blog post, originally published in 2021, has been updated for 2022.

How to Use Technology to Differentiate Instruction in the Classroom

Explore the challenges of differentiated instruction and how technology can help. Using technology to differentiate instruction assists in automating the process.

Kathleen Richards

December 29, 2023

10 Math Intervention Strategies for Struggling Students

These math intervention strategies for struggling students provide lessons, activities, and ideas to support Tier 1, Tier 2, and Tier 3 math students who are two or more years behind grade level.

Richard Blankman

May 7, 2024

How Differentiated Instruction Transformed a Charter School

Learn how Peak Charter Academy in North Carolina prioritized differentiation in the classroom, even when the pandemic hit the U.S.

April 7, 2021

Get free quick tips for bringing RTI into the core classroom.

• Professional Learning
• Intervention

Related Reading

Early reading intervention strategies for young learners

Shaped Contributor

August 5, 2024

How to create a classroom management plan

Jennifer Corujo Shaped Editor

July 30, 2024

AI for special education teachers

Dr. Suzanne Jimenez Director of Academic Planning and Data Analytics at HMH

July 26, 2024

36 easy math differentiation strategies for middle school

by Katrina | Jan 13, 2024 | Maths

When I began my journey as a science and math teacher, the idea of differentiating instruction seemed daunting. How could I possibly cater to the diverse needs of my students without drowning in multiple lesson plans?

The challenge was real, and panic set in.

How on earth was I meant to adjust all the different activities to meet all the different needs of my students? Did that mean I had to create 30 different lesson plans? How was I meant to do that as well as keep on top of my marking and assessment task writing and… you get the drift. I completely freaked out. 

It wasn’t until later in my career that I grasped the true meaning of differentiation and discovered manageable strategies for daily implementation.

That’s where this post comes in!

I’ve created a list of easy math differentiation strategies that are easy to implement on a daily basis and don’t require multiple lesson plans! 

So grab a coffee and sit back and relax while I give you a list of effective strategies to differentiate in your math classroom to make your job easier!

list of math differentiation strategies

Disclaimer: This blog post, ’36 easy math differentiation strategies for middle school’, may contain links to products I have developed.  Read full disclaimer here .  list of math differentiation strategies

What are math differentiation strategies?

Carol Ann Tomlinson defines differentiation strategies as the practices of proactive planning and inclusivity to ensure the learning experiences are accessible to all learners to meet their individual learning needs.

I love this definition as it really encompasses the main point – that differentiation strategies are used to meet the learning needs of students. 

Differentiation is an understanding of student learning needs and how to meet them. It requires successful incorporation of multiple strategies in order to meet the individual needs of those in your classroom. It  is not individualised learning and  does not require multiple lesson plans from you.

4 types of math differentiation strategies?

According to Tomlinson (2000), there are 4 ways you can incorporate math differentiation strategies in the classroom to foster learning opportunities.

1. Differentiate the Content

Differentiating the content means ensuring each student starts where they need to. Some students may need to start at an introductory level, while others can jump in at the extension questions. 

This can also include how students receive the content. For instance, whether they receive the content via the teacher, a video, visual resources, etc. 

2. Differentiate the Product

Math differentiation strategies relating to the product can refer to either the end product students produce to demonstrate their learning, or the standard of that product. 

3. Differentiate the Process

The process or method used is how students engage with the content. An example might be that you explicitly teach one group while having another do some research, or watch a video or do some hands-on modelling.

This also includes math differentiation strategies that make the learning accessible or achievable. An example may include providing more processing time for individual students. Another could be providing scaffolding that breaks down the concepts into manageable chunks.

4. Differentiate the Environment

The environment shapes how or where the activity is completed – this includes whether students complete the activity in groups or individually, and where they might complete that work in the classroom.

Now, let’s get onto the good stuff…

The ultimate list of math differentiation strategies 

36 easy math differentiation strategies in the classroom, list of math differentiation strategies: content.

Here are a list of various instruction strategies for differentiating the content in a math classroom.

1. Provide various entry levels

Some students might need instruction from the foundations of the topic. Others might need to explore the concept on a deeper level. Allowing variety in the entry point allows students to access the content at their level.

Here are some easy, low prep ways to do this practically:

• Divide your questions for a topic into sections so they gradually increase in difficulty. You can either let students choose which section they begin in or allocate those sections. Most math textbooks do this automatically. 
• Allow students to choose whether they listen to explicit instruction. After doing some pretesting it may become obvious that some students already have a good knowledge of the topic you are about to teach. 
• Provide the opportunity for peer teaching. If you have some students who already understand the concept then allow them to teach some of their peers. 
• In a textbook or worksheet with multiple questions, students have to get 3 answers correct in a row in each section before moving on to the next. This means those who have understood the content and are ready to move forward will be able to and it immediately differentiates the work for the entire class. 
• Change the grade level. Students may be studying shapes or equations, but perhaps you can assist those who aren’t ready by going back to the previous grade level content, or excelling by giving higher grade level examples. 

2. Incorporate videos

Videos are a great resource to use for math differentiation if teaching a mixed ability class. 

Instead of having the class watch the video together, allow students to watch it individually with headphones. This allows those who need to rewind and rewatch certain parts to do this. Another positive of this method is that students can do this more secretively – not feeling that they are holding the whole class up or making it obvious they are needing to spend more time on the learning part than others. 

Often I pair my videos with questions. My higher ability students are able to watch the whole video and then answer the questions. For my students needing more support, I give less questions and allow them to complete while watching the video and pausing when they understand the answer. Those needing even more support are provided with the approximate time in the video that the question is answered.

3. Jigsaw activities

Jigsaw is a way of grouping students. First, students are split into groups where, as a group, they are to research / investigate / learn about a specific part of the topic. 

For example, for studying angles on parallel lines in math, one group might study corresponding angles, another co-interior, another alternate etc. Once they have become ‘experts’ at their given topic they then get split into mixed groups where each student is considered the ‘expert’ of their own topic. In this group each student takes a turn to teach the group about their area. 

This can be done with random assignments of groups, or you can sort students into groups and provide the expert topic based on their learning needs. For example, alternate angles may be easier for students to grasp or calculate than co-interior. 

4. Incorporate student interests

Being able to know your students well enough to incorporate their interests can sometimes be overwhelming – particularly at the beginning of the school year. 

However, there are differentiation strategies you can use to do this without knowing all their individual likes, hobbies and sports. 

For example, knowing a lot of the students in my class play soccer, I try to use soccer themed examples when I’m teaching math topics like speed. 

5. Changing the context or application

The context or application of the learning can be differentiated. For example, one group of students may apply their learning to an everyday example, while another may apply it to an industrial example. 

6. Scaffolded notes

Scaffolded notes give students freedom to express their understanding while also being able to ‘doodle’ with diagrams, colouring or sketches. I like to use these super simple note-taking templates for topic summaries or while watching a video.

Click here to get them for FREE!

I actually find that my extension students often need this type of scaffolding to help become more concise in their notes. 

7. Encourage cross-curricular application

Some students might be ready to apply their knowledge across subject areas. By incorporating this type of learning, your extension or gifted students will be able to engage in critical thinking and higher order thinking skills. 

8. Less ‘drills’ and more problem solving 

If your pretesting shows that students already have a good knowledge base, allow them to skip the drills and launch straight into the application and problem solving questions. 

9. Have students write their own questions

Another way to extend students would be to ask them to write their own questions. This works well if you can pair up some of your extension students to work together. That way they can each write a question, have their peer complete it, then swap back again to mark their peer’s answer. The level of understanding and critical thinking required to write an appropriate question is far superior to that needed just to answer a question. However, all students can still engage at their own level.

Any math differentiation strategies that include students doing more of the work than you is a win! 

10. Graphic organisers / visual representation

Graphic organisers allow for the visual processing of concepts and ideas, and more specifically how they connect to other concepts and ideas. A way to differentiate using these is providing students who need extra support with a graphic organiser or a scaffolded graphic organiser, while those who need extension could create a graphic organiser. 

11. Task cards

Task cards are an easy activity to provide to the whole class. Why have I included it in my favourite differentiation strategies list?

Because they provide options of student choice. Choice in the order they complete the task cards and choice in how many they complete. 

12. Add personification

Personification is my favourite of all math differentiation strategies for engaging higher order thinking skills for students. This can be so easily added to any worksheet, activity, or task and super easy to add into a lesson if some students finish their work early. 

So what is it?

Personification is attributing human characteristics or personality to something that isn’t human.

Therefore, to incorporate personification into learning ask students to answer questions like these examples below:

• Angles: what would a co-interior angle say to a corresponding angle?
• Shapes: What would a square say to a triangle?
• Order of operations: What would a division symbol say to an addition symbol?
• Pythagoras: What would a hypotenuse say to a right angle?

Want a free printable version?

Enter your email address here and I will send you a pdf printable version of the 36 Easy Math Differentiation Strategies for your classroom. 

This is perfect for adding to your teacher planner or having up next to your desk for your next lesson planning session.

(unsubscribe at any time)

Please check your email to download your printable pdf.

List of math differentiation strategies: Product

13. offer choice for the type of activity or type of product.

I used to think this was so much work as I didn’t want to have to make 4 different lessons for students to choose. 

But you don’t have to do this! 

For example, math summary tasks are easy to differentiate in this way as students could choose how their final product will look. Will it be a brochure? A poster? Video? Slideshow? You can still provide the same success criteria and have students cover the same key points, but give them choice in how they would like to present it.

For activities, rather than having students go through all stations set up around a room, give them a number to complete. If you set up 5 stations then ask students to choose three to complete. This also allows those who may finish sooner to have the opportunity to complete an extra station. It also means that for those students who need extra time you could easily differentiate and ask them to only choose two to complete.

14. Differentiate the success criteria

While you might be providing students the same activity to complete, differentiation could come in with the success criteria you provide for students. Students who need some extension could have different levels of success criteria to meet. These could vary in terms of depth or breadth of understanding shown, or the quality of product produced.

For example: Pythagoras’ theorem – same worksheet of triangles:

Success Criteria Level 1: students can highlight the hypotenuse of a right angled triangle and state pythagoras’ theorem.

Success Crtieria Level 2: students can calculate the length of the hypotenuse when given the other two sides of a right angled triangle.

Success Crtieria Level 3: students can calculate the length of a shorter side when given the other two sides of a right angled triangle.

15. Change the verb

Differentiating the verb used can prompt students to deliver various products. E.g. design, create, evaluate, assess, compare etc. Blooms taxonomy can be a helpful reference for this.

16. Allow for the expression of creativity

Allow students to be creative with the end product. This could be done by offering choice for presenting information via a model, diorama, painting, sculpture, drama, song etc. This is also one of the low prep math differentiation strategies for you as it leans on student agency rather than your creation of new material.

17. Interview students

I’ve often come across student’s who struggle to express their level of understanding on paper, but can very clearly express it verbally. This is a great option for informal assessment and can be done during a regular class lesson. Math differentiation strategies within interviewing could include the types of questions asked, the phrasing of the questions, the format you want them to answer in, the length of time given for an answer.

differentiation is the difference between busy work and learning #education #edchat #differentiation — The Animated Teacher (@katrina_harte) November 2, 2021

List of math differentiation strategies: Process

18. use technology.

There are lots of different programs that promote easy math differentiation strategies by offering students choice. For example, Quizlet allows students to choose how to learn the content. Choices include flashcards, multiple-choice questions, typing an answer, practicing spelling, matching the correct term to definition, or playing a game. 

Differentiation using technology could also be offering choice in the type of technology used. Students could choose whether to use a tablet, computer, phone or alternative.

19. Offer choice for the order they complete tasks

While there is often a need to have students complete tasks in a particular order, be aware of opportunities to change up that order. Allowing students to choose their own adventure allows students to learn the material in the order that makes sense for them. 

20. Use stations 

Stations allow for students to move around the room and often choose who they are working with and how long to spend on each activity. I will often use normal classroom activities, task cards or even just worksheets, and place them around the room and call them ‘stations’. 

Students have a choice of where to start, who to work with, and how to stand / sit / group around the station. 

This also allows you to tell students who might need extra time to complete less stations.

21. Cut and paste activities 

Allowing for students to physically manipulate something can be so powerful for those needing math adjustments. It helps their brain to process the information in a new way. This can be easily done in class with simple worksheets. For example, if you are wanting students to match the term with the glossary definition then provide students with a printable version they could cut and paste.

 This offers another opportunity to provide choice as students could choose to cut and paste, or use colour coding, or write the term in the box with the definition. Three different options for one activity and no extra prep from you!

In maths this could even be a worksheet of equations and the answers that students cut and paste or manipulate to match. 

22. Hands-on learning

Providing students with the opportunity to explore learning in a hands-on way provides immediate differentiation as students will engage in a way that makes sense to them. 

To differentiate you could provide multiple types of materials for students to choose from. For example, to learn about ratios in maths I provided both cordial and paint for students to explore. 

I always try to use manipulatives as much as possible in my math lessons as it helps students visualise concepts in a way just reading or writing cannot. 

23. Modelling

Modelling can be done in many different ways. It could involve going through step by step showing students exactly how you want them to go about solving a problem. This could be done while sending your extension students off to investigate the topic. Modelling a process could also be done by providing a scaffolded worksheet for students who need it.

24. Provide written or printed instructions broken down into steps

This is one of those math differentiation strategies that you can do for the whole class rather than just a few students. It won’t hinder the rest of the class to have instructions broken down into steps. It is also important for students to have these visually represented to them so they can refer back as often as needed, whether it be printed or upon the board.

This is particularly helpful when getting to higher levels of math that require multiple steps to solve a problem or an equation. Breaking it down into steps and having it available to students who need it is a great way to differentiate in math.

25. Change the reading level

It is important that students are given the opportunity to engage in learning by being provided with resources that are at an appropriate reading level. Use programs such as ChatGPT to rewrite passages and activities at appropriate reading levels. 

26. Provide extra processing time

This could be as simple as giving students fewer questions to complete in the same amount of time. 

27. Provide class discussion questions before discussion time

This allows students who need extra processing time to have the opportunity to still be a part of a class discussion. 

These could be a homework task or as simple as handing out the questions before marking the roll so the students have time to read the questions in advance. For some students, this could be the difference between being able to contribute to a class discussion, or not. 

28. Give warning before being called upon in class

Following on from the previous point, students may freeze or shut down when being called upon in class if they haven’t had a chance to consider and process the question. 

So, if the activity is to answer a few questions and then go through them as a class, you could go and quietly say to the student that you are going to ask their opinion about question #3. This gives them time to process it, time to ask you questions if they don’t understand, and time to make it an answer they are proud of. 

We need to keep students engaged, not busy. — Brian Aspinall (@mraspinall) October 11, 2021

list of differentiation strategies

List of math differentiation strategies: Environment

29. group work.

Using multiple forms of grouping for students in a class provides differentiation as students take on different roles within their groups depending on who they are with. Some examples of flexible grouping could be:

• Grouping students who need some extra support together. Ability groups allow you to provide this group with more explicit instruction as you move around the room.
• Grouping students of mixed ability together. This allows those who need extension to take on a leadership role within the group and have the opportunity to share their understanding with their peers.
• Groups based on choice. This could be student choice for who is in their group, or students could be grouped by their choice of activity. Both of these options allow for differentiated instruction and learning. 
• Group in different group sizes: some students might need small groups while others can work in larger groups to complete the same task.

30. Offer choice for how they work

Offering student choice leads to increased engagement as students feel they have ownership over their learning. Allowing students to choose how they work, whether it is individually, with a partner, as a small group, etc is an easy way to incorporate math differentiation strategies into your classroom without loads of preparation. 

31. Where they complete the work

Allow students to choose whether to stand, use different chairs, sit on the floor, work outside etc. Taking a class outside for a lesson on the lawn is fantastic for this. Since there are no chairs, students can choose whether to sit, stand, lie on their stomachs, sit on a rock, choose to sit in the sun or the shade. So much choice!

32. Brain breaks

Brain breaks are so important for retaining high levels of student concentration when learning new concepts. Providing choice in how they have breaks enhances your differentiated classroom. 

33. Allow to complete work in a small group

Allowing some students to work in a small group as opposed to completing a task individually can be a good differentiation option. This provides the support of their peers and together they may be able to accomplish something that individually they wouldn’t have been able to.

34. Change the learning environment 

Changing the environment for students can be very powerful. This can include allowing for some students to sit in a more quiet space, while others can work in pairs. This could also include where the students’ desks are facing. One student may learn more effectively with their desk at the front of the room facing the board, while another can work opposite a peer.

35. Allow students to remove themselves from distraction

Similarly to above, this refers to allowing choice for students. For example, wearing noise-cancelling headphones or the freedom to move around the class if needed.  Allowing this freedom can allow students to take ownership over their learning and concentration by being able to change their environment if needed.

36. Allow gifted students to work together or with students from higher grade levels 

This isn’t always the easiest to organise but if there is an opportunity it can be an invaluable experience for students in both grades.

For instance, I once had a student in year 10 who was super passionate about biology and DNA. We organised for him to go and teach a lesson to the year 12 biology students. Needless to say, he was so chuffed and worked so hard to prepare for the lesson. The year 12 students were definitely taken aback by his depth of knowledge and understanding and it inspired them to take their learning beyond the curriculum too.

Share the love!

Don’t keep this list of math differentiation strategies to yourself! Share with your teaching friends!

Did you find this helpful? Got some of your own awesome math differentiation strategies?

Please comment below!

• Tomlinson, C. A., 2000. Differentiation of Instruction in the Elementary Grades . ERIC Digest . ERIC Clearinghouse on Elementary and Early Childhood Education.  

Written by Katrina

Recent posts.

Check out my best selling no-prep lesson activities!

We noticed you're visiting from Australia. We've updated our prices to Australian dollar for your shopping convenience. Use United States (US) dollar instead. Dismiss

Differentiation

Related Topics: A Level Maths Math Worksheets

Examples, solutions, videos, activities and worksheets that are suitable for A Level Maths to help students learn how to differentiate equations.

A-Level Maths : Differentiation 1 In this tutorial you are shown how to differentiate terms of the form ax n where n is a positive integer

A-Level Maths : Differentiation 3 In this tutorial you are shown how to differentiate fractional equations that will reduce to the form ax n for each term.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.

Efficient numerical algorithms for solving a time-fractional diffusion equation with weakly singular solution

New citation alert added.

This alert has been successfully added and will be sent to:

You will be notified whenever a record that you have chosen has been cited.

To manage your alert preferences, click on the button below.

New Citation Alert!

Please log in to your account

Information & Contributors

Bibliometrics & citations, view options, recommendations, finite difference methods for the time fractional diffusion equation on non-uniform meshes.

Since fractional derivatives are integrals with weakly singular kernel, the discretization on the uniform mesh may lead to poor accuracy. The finite difference approximation of Caputo derivative on non-uniform meshes is investigated in this paper. The ...

Finite difference/spectral approximations for the time-fractional diffusion equation

In this paper, we consider the numerical resolution of a time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative (of order @a, with 0=<@a=<1). ...

A unified numerical scheme for the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients

We consider the numerical solutions of the multi-term time fractional diffusion and diffusion-wave equations with variable coefficients in a bounded domain. The time fractional derivatives are described in the Caputo sense. A unified numerical scheme ...

Information

Published in.

Elsevier Science Publishers B. V.

Netherlands

Publication History

Author tags.

• Fractional diffusion equation
• Weak singularity
• Graded mesh
• Adaptive mesh
• Convergence
• Difference scheme
• Research-article

Contributors

Other metrics, bibliometrics, article metrics.

• 0 Total Citations
• 0 Total Downloads
• Downloads (Last 12 months) 0
• Downloads (Last 6 weeks) 0

View options

Login options.

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Share this publication link.

Copying failed.

Share on social media

Affiliations, export citations.

• Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download
• Download citation
• Copy citation

We are preparing your search results for download ...

We will inform you here when the file is ready.

Your file of search results citations is now ready.

Your search export query has expired. Please try again.