Challenging Concept: Integers

I teach grade 6s and 7s and integers is a common concept in math my students struggle with. Particularly, when dealing with negative integers. This includes adding, subtracting, multiplying and dividing with negative integers. Specifically, my students have had difficulties with understanding that adding a negative integer makes a number less positive and that subtracting a negative integer makes a number more positive. Though we go over the rules of integers, I have seen significant students experience difficulty with the concept. I have also utilized metaphors such as thinking of negative integers as “unhappy things” and positive integers as “happy things” and if we add more positive integers, we will be more happy and the number will be more positive and vice versa. However, if I subtract a negative number, I am metaphorically speaking taking away unhappy things, and therefore I will be more happy and the number will be more positive.

3 Step T-GEM cycle

Teacher Strategies Examples Student Strategies
Provide background information on integers Introducing what “positive integers” and “negative integers” look like
Generate Show examples of different types of integer equations, but starting only with adding of two positive and two negative integers. (+2) + (+3) = +5
(-2) + (-3) = -5
Try to generate relationship between positive and negative integers and operation. They also try to consider how this math concept is used in real life applications.
Evaluate Encourage students to evaluate their relationships to see if the integer equations will become true/false. “What are some other examples?”
“Create your own examples and see if it follows your rules.”
Try out their theories and evaluate them.
Modify Ask students to modify original ideas of relationship between positive integers.

Then, the teacher will introduce a new related concept such as adding a negative integer, then subtracting a negative integer, before moving on to multiplying and dividing.

“What changes can we make to your rule?” Modify their relationships if it is false.


Digital Technology

A digital technology that can be used to accompany the concept of integers is the use of coloured chips found at http://nlvm.usu.edu/en/nav/frames_asid_161_g_2_t_1.html?from=search.html and http://nlvm.usu.edu/en/nav/frames_asid_162_g_3_t_1.html?from=search.html. They allow students to visually represent the integers using different coloured chips (e.g. one for negative, one for positive).

Index of Virtual Manipulatives. (2017). National Library of Virtual Manipulatives. Retrieved 23 February 2017, from http://nlvm.usu.edu/en/nav/search.html


  1. Hi Gloria, I can only imagine how conceptually challenging this would be for many students. In high school, so much of these types of misconceptions are masked because of access to calculators. I think if I taught Math 9, that I would be tempted to give a diagnostic test on skills such as integers, fractions, one-step algebra… the basics. My guess is that I would be shocked to know how many Grade 9s still don’t have integers down pat. ~~~In other thoughts…. In my very first year of teaching, I had a remedial Math 8 class. I remember that these kids (not math lovers, to say the least) enjoyed the human number line activity. I taped a massive number line down the hallway, and they had to perform operations with their entire body! I wonder if you had a group of “happy” kids and “unhappy” kids at either end, if that would help reinforce your analogy? Another analogy, well, actually “real life” application that seems to help my Grade 10s, is to add dollar signs to the numbers. Humans seems to be genetically wired to understand when somebody owes them money or if they have won money from being in a deficit. Anyhoo, I love the fact that you took on a math topic for this. Great job wrapping GEM around it!!! ~Dana

    1. Hi Dana,

      Thank you for the feedback and for the insight! I like the number line activity that encourages students to use their entire body because it is a full on manipulative! I also will utilize more practical aspects such as the borrowing money concept!

  2. Hi Gloria!
    I agree with Dana on that integers is definitely a challenging topic to teach new and you’re doing a great job with your happy/unhappy people analogy. The kinesthetic element of a human body number line would be such a blast in my opinion. In grade 11 and 12 Physics, integer knowledge is taken for granted so when I come across students that are still unclear of integer operations, I would be very much inclined to incorporate the ideas you have put forward for review purposes.

    Thanks for sharing

  3. Hi Gloria,

    In the evaluation phase, could you provide an example based on your experience teaching this concept of what students’ ideas might entail (see try out their theories part)? “What are some other examples?” “Create your own examples and see if it follows your rules.” Try out their theories and evaluate them. Also, in what way might the digital technology of the colored chips assist with evaluation? Thank you for providing the link and your ideas on teaching this challenging concept by encouraging students to generate the rules in math, Samia

    1. Hi Samia,

      Thanks for the questions! I actually have not tried to implement this method of teaching integers using the T-GEM framework. However, I envision that students will be trying to add a negative integer and a positive integer as if both integers are positive or subtract a negative integer from a positive integer as if both integers are positive. The digital chips will help them visualize what it means to subtract/add negative and positive answers. Students can also use the chips to demonstrate their theories on the board to their peers. With a partner, peers can also work together to confirm or nullify their theories.

  4. Hi Gloria,

    Well done on taking on integers for this GEM assignment – definitely a tricky concept for students to understand! I have taught adapted math from grades 8-11, as well as elementary math, and I can say that in every class at every level, integers was a difficult concept for students. I really liked how you created a scenario that students could relate to with “happy things” and “unhappy things” that students could add or subtract. I especially liked your explanation that “if I subtract a negative number, I am metaphorically speaking taking away unhappy things, and therefore I will be more happy and the number will be more positive.” This is a very creative way to help students understand how to relate to negative/positive numbers. As I live in an area where temperatures can vary between -30 and +30 degrees Celsius, depending on the time of year, I often use temperature to teach students about integers. Having said that, the subtraction of a negative continues to be a struggle as this concept does not work well based on a thermometer analogy and does not provide a clear relationship for students – I like your model better as it encompasses all aspects of integers. I also like that you have asked students to create their own examples of situations using positive and negative numbers which encourages them to think more deeply about the concept, but also to apply it to a scenario that interests them or that they can connect to. I would be interested in hearing some of the scenarios that your students came up with. Do you tend to have many students who need quite a bit of guidance during this activity?

    1. Hi Mary,

      I like the use of the current climate of your teaching environment to teach integers. It’s authentic and relevant!

      I have not utilized T-GEM to teach integers to my students, but from my experience there is a big range of levels of understanding on this topic. Some can grasp it without the visuals or metaphors. Others can utilize the metaphor really well with the assistance of visuals (e.g. drawing happy and sad faces and taking away or adding on). But there are some who still continue to struggle. My hope is that T-GEM encourages them to explore how integers work rather than directly teaching them explicitly.

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