In working with students who have fallen significantly behind in math, a common challenge is adding and subtracting. This can manifest in word problems or simple add and subtract problems. When this is the case, the first thing I check is whether the student understands conceptually what addition and subtraction are. Here is a Google Slides file that shows the approach I use. (You can make a copy and then edit.) I copy and paste slides for subsequent days so the Google Slides file services as a repository of the trials – data collection.

Each slide as the same format.

The operation in bold font.

The primary pile with the objects to work with.

The secondary pile with objects used for addition prompts.

The garbage can for objects discarded in subtraction prompts.

I like to use pennies as objects but will use Google Images of topics the student likes if I need something extra to keep their interest and attention. The student is tasked with performing an action with the objects and to distinguish between tasks to show discernment of what action to perform, relative to the prompt. There are 3 operations types: addition, subtraction, and sorting. The sorting is used simply as a distractor. Each image shows the original slide and a slide of the final product. The slides can be copied to for additional prompts. This first round focuses on common language that speaks to addition and subtraction.

In the next round, the sorting is removed and the language is focused on the terms add and subtract as a step in shaping understanding of the eventual symbols.

Finally, the actual symbols are used. If the student gets confused the previous language can be used as a prompt.

We explain steps in great detail to students but often omit the underlying concept. The topic of adding or subtracting fractions with unlike denominators is an example of this.

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The example above right is a short cut for what is shown above left. These short cuts, which math teachers love to use, add to the student’s confusion because these rules require the student to use rote memorization which does is not readily retained in the brain.

I suggest using what I call a meaning making approach. I present the student 2 slices of pizza (images courtesy of Pizza Fractions Game) and explain the following setting. “You and I both paid for pizza and this (below) is what we have left. You can have the pizza slice on the left and I will have the pizza slice on the right. Is that OK?” The student intuitively understands that it is not because the slices are different sizes. I then explain that when we add fractions we are adding pizza slices so the slices need to be the same.

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I then cut the half slice into fourths and explain that all the slices are the same size so we can now add them. Then the multiplying the top and bottom by 2 makes more sense.

The use of the “-” symbol is challenging for many students. They don’t understand the difference between the use of the symbol in -3 vs 5 – 3. To address this I use a real life example of multiple uses of the same symbol (1st 2 photos below) then break down the “-” symbol (photo below at bottom). I suggest this be introduced immediately prior to the introduction of negative numbers.

From the blog Bits and pieces. This would be a great poster to put on a learning wall while kids are learning the rules. In lieu of showing the kids repeatedly what to do simply have them follow this – training to follow examples.