Category Archives: Number Sense

Adding ones digits in 2 digit numbers with carrying

A major obstacle in math for many students with special needs is carrying in addition problems. Below is a task analysis approach.

First, I target the step of identifying the ONES and TENS place in the 2 digit sum in the ONES column (below it is 12). In a scaffolded handout I create a box to for the sum with the ONES and TENS separated. At first I give the sum and simply have the student carry the one.

sum of ones given.jpg

Then I have the student find the sum and write it in the box (14 below). Once mastered I have the student write the sum and carry the 1.

sum of ones given

They would have mastered adding single digit numbers before this lesson. I revert back to single digit numbers to allow them to get comfortable with writing the sum off to the side without the scaffolding. (In the example below I modeled this by writing 13.)

sum of ones with color no scaffolding

The last step is to add the TENS digits with the carried 1. I use Base 10 manipulatives to work through all the steps (larger space on the right is for the manipulatives) and have the student write out each step as it is completed with the manipulatives.

sum of ones with carrying with base 10 blocks first

Finally, the student attempts to add without the scaffolding. I continue with color but then fade it.

adding 2 digit numbers with carrying with color no scaffold

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Place Value Representation

Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.

pace value representation just numbers.jpg

The focus can shift to money as well.

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The I in Instruction can be the same I in IEP and IDEA.

I am consistently surprised by the reliance on canned items for students who struggle. There are different reasons students struggle but we know that there are secondary characteristics and factors that inhibit effective information processing that can be addressed with some Individualization.

In a math intervention graduate course I teach at the University of Saint Joseph, my graduate students are matched up with a K-12 student with special needs. The graduate student implements instructional strategies learned from our course work. Below is the work of one of my grad students. From class work and our collaboration we developed the idea of using the fish and a pond as base 10 blocks for the student my grad student was helping. He likes fish and fish will get his attention. The grad student explained that if he has 10 fish the 10 fish go into a pond. In the photo below the student modeled 16 with a TEN (pond) and 6 ONES (fish).

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Similarly, another student likes Starbursts and that student’s respective grad student created Starburst packs to represent TENS and ONES (there are actually 12 pieces in a pack so we fudged a little).  The point is that it was intuitive and relevant for the student. The student understood opening a pack to get a Starburst piece.

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More Than: Concept and Symbol

The alligator eats the bigger number is the common approach for student to use inequality symbols (<  > <  > ). I find that students remember the sentence but many do not retain the concept or use the symbols correctly, even in high school. The reason, I believe, is that we introduce additional extraneous information: the act of eating, the mouth which is supposed to translate into a symbol, the alligator itself. For a student with processing or working memory challenges this additional information can be counter productive.

alligator eats bigger number

I use the dot method. By way of example here is the dot method. I show the symbols and highlight the end points to show one side has 2 dots and the other, 1.


Then I show 2 numbers such as 3 and 5 and ask “which is bigger?” In most cases the student indicates 5. I explain that because 5 is bigger it gets the 2 dots and then the 3 gets the 1 dot.

3 dots 5

I then draw the lines to reveal the symbol. This method explicitly highlights the features of the symbol so the symbol can be more effectively interpreted.

3 less than 5.

That is the presentation of the symbol. To address the concept of more, especially for students more severely impacted by a disability, I use the following approach. I ask the parent for a favorite food item of the student, e.g. chicken nuggets. I then show two choices (pretend the nuggets look exactly the same) and prompt the student to make a selection. This brings in their intuitive understanding of more.

concept of more chicken nuggets just plates

I think use the term “more than” by pointing to the plate with more and explain “this plate has more than this other plate.” I go on to use the quantities.

concept of more chicken nuggets more than words

Finally, I introduce the symbol to represent this situation.

concept of more chicken nuggets more than symbol

Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!

Lucas less than example

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Opportunities for Parents to Engage Students with Math

Math is often considered an esoteric set of information that is disjointed from the reality people face, aside perhaps from money. Sadly, in school, especially in older grade levels, math is indeed presented this way.

sally math problem

A situation as simple as riding an elevator provides opportunities to show and engage a student with math applied in authentic and common situations. For example, the elevator buttons address counting and cardinality (4 indicates a total of 4 floors – ignoring the R), comparison (if we are on floor 2 and need to go down, which floor do we go to?) and measurement (height above ground floor measured in floors). Such situations also provides opportunities for generalization into other settings – the important settings of every day life!

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Authentic Activities – Money and Prices

Below is a photo of a typical worksheet for money. I worked with a parent of a high school student severely impacted by autism and she explained that her son worked on nothing but worksheets when he worked on math. For students with more severe disabilities the worksheet is not real or meaningful. The photos and the setting is abstract.


Below is a photo of shelves in a mock grocery store we set up at our school for students who were in a life skills program. They would have a shopping list, collect the items in a basket then compute the total cost. We had a mock register set up (eventually we procured an actual working register) and the students made the same types of calculations they would on a worksheet but in an authentic setting, which was more concrete. We would start with simple money amounts, e.g. $1.00 then make the prices increasingly more challenging, e.g. $1.73.

mock grocery store

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Trick for subtracting integers

I have algebra students, in high school and college, who struggle with evaluating expressions like 2 – 5.  This is a ubiquitous problem.

2-5 problem

I have tried several strategies and the one that is easily the most effective is shown below. When a student is stuck on 2 – 5 the following routine plays out like this consistently.

  • Me: “What is 5 – 2?”
  • Student pauses for a moment, “3”
  • Me: “So what is 2 – 5?”
  • Student pauses, “-3?”
  • Me: Yes!


I implemented this approach because it ties into their prior knowledge of 5 – 2. It also prompts them to analyze the situation – do some thinking.

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1 to 1 Correspondence Scaffolded

IMG_20171221_080306139 - Edited

I have encountered several students who struggle with 1 to 1 correspondence with the educators struggling to figure out how to teach this to these students who continue to struggle. This post reveals an approach I used with a student.

I broke down the task using a formal task analysis approach. This approach involves identifying the different individual steps and to address these steps in isolation. Here is the sequence I use and suggest.

  1. Conduct a pretest using a task analysis pretest data sheet I created for this topic. I do not use any scaffolding and prompt the student to count out the objects (in this case decks of cards) and to do so independently. I prompt the student after they show they cannot complete a step which allows the student to attempt the next step. (Think of teaching a student to get dressed and he cannot put his socks on. You help him with the socks then ask him to put on his shoes.)
  2. I then focus on the movement of the objects. I provide scaffolding for start and stop piles (see mats with track photos above). The student is asked to move the cards one at a time without counting.
  3. The student must learn the “rules of the game” which includes how to place the items in the stop pile. Students may be confused about placement, e.g. one student ran out of room while placing the decks in a straight line and I had to demonstrate that it was OK to place them on different spots on the mat. Once the student demonstrates mastery of moving the items we move on to the next step. IMG_20171221_080726834 - Edited
  4. We then focus on counting in isolation. The card decks are labeled with numbers (photo below) and the student does not move anything but simply reads the numbers. (More on these numbers in a later step.) More numbers can be added as necessary.IMG_20171221_080604425 - Edited
  5. The next step (photo below) is to have the student read the number on each card. I have a stack of decks of cars on the start pile with the numbers facing down. I show the student the number of the deck that I am moving to the stop pile and the student reads off the number. I place the used deck face down to hide the number. This activity forces the students to focus on each item as he reads the number. One student kept counting ahead to the next number and I prompted him to return his focus to the current number. This is the crucial step as it focuses on the 1 item 1 number aspect of counting. IMG_20171221_080526960 - Edited
  6. The next step is to have the student move the decks from the start pile to the stop pile and to read each number while doing so. I turn each deck face up as a prompt for the student to move and read.
  7. The student then is prompted to select the cards on his own and read (the cards can be in a pile in order by number).
  8. Eventually 1 then 2 then 3 decks have the number missing which adds an extra task demand for the student – identify the next number as he is moving the item.
  9. Finally the items do not have any numbers and the student counts, with the mats eventually be faded.

Note: this is especially effective for students with ADHD because it helps to focus and organize their task demand for the activity of counting.

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