Elapsed Time Scaffolded

Elapsed time scaffolded

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The final answer would be 4 hours and 25 minutes.

Elapsed Time Scaffolded and completed

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AWESOME Online Graphing Calculator

Desmos graph pizza function

This can be a game changer for students with special needs who struggle with math. The Desmos graphic calculator allows students to interact with math equations through multiple representations. It is far superior to graphing calculators in terms of quality and ease of use and is free. The app for Smartphones is outstanding.

Here are features that make this calculator user-friendly and an outstanding instructional strategy.

  • Students can click on dots and the ordered pair will appear (see top photo below).
  • Students can change features of the equation and immediately see how the graph changes.
  • Students can collect data and create a graph and convert the data into “mathy” representations like equations (see top photo below).

Desmos regression price vs mileage of used cars


Desmos parabola



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Math Rulz

Screenshot 2017-03-23 at 11.45.11 AM

Math is challenging for various reasons:

  • It involves numbers and not just words that we use on a regular basis to communicate.
  • It is a language all of its own so students have to learn the language as well as the concepts.
  • We teachers sometimes make learning the math more challenging.

I want to elaborate on this last reason. The photo above speaks to this. We present a topic. If the students struggle there is a tendency to “dumb down” the topic to rote memorization of a meaningless set of steps. Below is 1 of dozens of examples of memorization tricks we use. I call these Math Rulz.

Integer opreations for Math Rulz

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Fractions! Meaning Making for Comparing Fractions


Fractions is one of the most challenging math topics. Many high school and college students struggle to some degree with fractions.  The Common Core of State Standards (CCSS), despite all the criticism, includes components to address the conceptual understanding of fractions. Below is a photo showing a 4th grade Common Core standard regarding fractions along with an objective for a class lesson I taught at an elementary school in my district. I subsequently presented on this at the national CEC conference in 2014. Notice the bold font at the bottom, ¨justify…using a visual fraction model.¨ The photo above shows an example of a model I used in class.

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The photo below shows a handout I used in the lesson. The first activity involved having students create a Lego representation of given fractions. These would eventually lead to the photo at the top with students comparing fractions using Legos. The students were to create the Lego model, draw a picture version of the model then show my co-teacher or I so we could sign off to indicate the student had created the Lego model.

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The Lego model is the concrete representation in CRA. In this lesson I subsequently had students use fractions trips (on a handout) and then number lines – see photos below.

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RTI – Response to Intervention

RTI Process

The photo above shows a model of the RTI (called SRBI in Connecticut) process. RTI is a systematic approach to addressing student academic needs. Here is a link to a video explaining the process and below is an outline of the process:

  1. Students are served in a classroom that provides high quality initial instruction. This includes the use of UDL, differentiation, formative assessment, instructional strategies to make content meaningful and concrete and to meet student needs in general. The general classroom is Tier I.
  2. Assessment is used to evaluate student progress AND the effectiveness of the instruction. If students are not understanding a math topic or unit (as demonstrated by data not observation) the student can be moved into Tier II which involves intensified focus of instruction and in a small group.
  3. Assessment is used again. If the student is not making sufficient progress despite changes in instruction the student can be moved into Tier III which involves maybe 1 on 1 or 1 teacher and 2 students. The level of intensity is ramped up further.

Here are a couple of key components:

  • The initial classroom includes an effort to meet individual needs.
  • Data is the key to decision-making. Assessment is objective.
  • The programming is evaluated using the data.
  • RTI is included in IDEA 2004.
  • Student placement at the different tiers is fluid. Students are moved into and out of tiers based on data.


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Snow Math

2017-03-14 10.03.23

Here’s a common word problem used for linear functions and equations (y=mx+b):

There are 6 inches of snow on the ground. Snow is falling at a rate of 2 inches per hour. Write a linear equation showing total snow as a function of time (in hours). The equation would be y=2x + 6.

Often the word problems like this are presented on a sheet of paper in isolation as an attempt to make the math relevant and to develop conceptual understanding. For students who have trouble with conceptual understanding, words on paper are likely too abstract or symbolic to allow applications like the one above to be meaningful.

The real life application is useful if presented more effectively. Here’s an approach to use the same scenario but in a more relevant and meaningful presentation. The photo above shows the current amount of snow – call it 6 inches. Students can be shown the photo to allow for a discussion about accumulation and for their estimates of the amount of snow shown. The photo below shows an excerpt from a storm warning. Showing this warning and a snow fall video can allow for a discussion about rate of snow fall and the purpose for storm warnings. Combined, this approach can lead into the above word problem.

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Once the application is presented students can be asked to compute snow levels after 1 hour, 2 hours etc. Then they can be asked to determine how long it would take for the accumulation to reach 18 inches (the prediction for the day this post was composed). After computing the answers WITHOUT the equation the students can be shown how to use the equation – the “mathy way.”

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Zone of Proximal Development


The photo above shows 3 levels of task demands for children based on Vygotsky’s levels of development.

  • On the left is a level in which the student can readily perform the task independently, i.e. he is doing something he already knows how to do.
  • On the right is a level that is too challenging for the student to accomplish independently. It is something he cannot do and does not know how to do.
  • In the middle is a sweet spot. The level involves tasks that are accessible to the student but with support – scaffolding.

In reading this is known as the “instructional level” – see photo below. Reading material is evaluated by determining how challenging it is for a student. Material that the student can read independently allows for some growth in reading ability. Material that the student finds too challenging would not allow for substantive growth. In the middle is the sweet spot – the Zone of Proximal Development.

Instructional Level

We can do the same with math using scaffolding. In the photo below is work performed by a former 7th grade student of mine with Asperger’s who tested at a 1st grade math level. I used colored pencils and 2 sided tokens to support his work with integers (red for negative and yellow for positive) in a CRA approach. The color coding and tokens were like the swimmies in the photo above of the child in the ZPD. Eventually these supports were faded. Throughout this process I was constantly pressing him to do more with a little less assistance.

adding integers chips and colored pencils

I want to emphasize 2 major points regarding this.

  • Substantive learning occurs when a student has to step beyond his or her current ability level – the ZPD.
  • Often in schools educators avoid this, especially for students with special needs, because we want students to be engaged and successful (in the short term). We often confuse being active with learning. The guy on the tricycle in the top photo was performing a task but was he learning? (Note: this is not a student with special needs but a guy having some fun.)

Here’s are a video that fleshes out this idea.

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Factoring Numbers


Factoring, as used in elementary school,  is the act of changing a number into numbers that multiply to produce the original number. For example, 12 can be factored to 3×4. 3 and 4 are called factors of 12.

In the photo above, a “factor tree” is used to help identify the factors. I have often seen the factor tree used as the initial approach to teaching factors. I’ve also seen it used as the primary means of providing intervention for students struggling with factoring.  Think about that. Students who didn’t understand the initial instruction that likely involved the mathy approach shown above were provided the SAME approach.

Math topics can be presented with a more concrete introduction which can allow for more in-depth understanding. The photo below shows a CRA approach to factoring. This approach can be used as part of UDL or as an instructional strategy for intervention.





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FRACTIONS! Meaning Making for Adding Fractions


Fractions is one of the most challenging topics in math. Here’s an approach to help introduce fractions.

I show the photo above, explain to a student that he and I both paid for the pizza. We are going to finish eating the pizza and I get the slice on the left. I ask “is this fair?” This leads into a discussion about the size of the slices and what 1/2 and 1/4 mean. The pizza on the left was originally cut into 2 slices so the SIZE of the slices is halves. The SIZE of the slices in the one on the right is fourths. I have 1 slice left and it is a half so my pizza is 1 half or 1/2. He has 1 slice left and it is a fourth so his pizza is 1/4.  The bottom number is the size and the top number is the # of slices.

We cannot count the number of slices because they are not the same size. So we need to change my pizza.  So I slice my pizza and now I have 2 slices and they are cut into fourths. So now I have 2/4.  Note: I don’t show the actual multiplication to show how I got the 2 and 4.  I am sticking with the visual approach to develop meaning before showing the “mathy” approach.


Now that I have slices that are all the same size, I can now count the # of slices. “1, 2, 3…3 slices and they are cut in fourths.”


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Corresponding Angles in Stain Glass


Found this (above) cool example of corresponding angles (see photo below for explanation). This window photo could be a nice introduction to this type of problem by printing it out on paper and having students match angles as the teacher shows the photo on the Smart Board or screen.


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