Excellent production and effective explanation.

Graphing linear functions and the underlying concept are challenging for many students. The video below shows a scaffolded approach to teaching how to graph. This approach also addresses the concept of the graph as a visual representation of all possible solutions (see photo above). Students often do not realize that the line is actually comprised of an infinite set of points which represent all the solutions. Here is a link to the document used in the video.

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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.

- 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.)
- 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. - 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.
- 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.
- 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.** - 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.
- 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).
- 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.
- 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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A Kahoot is an online and app quiz game that allows students to answer questions using a personal device (e.g. simulated phone in photo above). The teacher can create the questions (e.g. example question I created in photo above).

My approach is to use a Kahoot to scaffold learning. In this post I use plotting points as an example.

- I start with simple questions, e.g. identify the letter and number coordinates for the dog and chick below. Notice in the top photo below that I provide the actual coordinates in question 1 (“for the dog C4”) as a scaffold to show the students what to do.
- Then I show numeric coordinates for a point, but only with positive numbers.
- Eventually I present problems that address all 4 quadrants and ask questions about the parts of the coordinate plane (photo bottom one, below).
- Notice that the questions have times (in seconds). This indicates the time allotted to answer each question (teacher sets this). For students with special needs I print a hard copy to allow them more time to read the question. If necessary, they can respond by circling the answer on the handout.

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As Piaget highlighted, our brains make connections between new information and previous information (prior knowledge). I introduce the concept of congruent triangles by connecting it to prior knowledge of identical twins (photo above).

This connection is carried throughout the chapter. For example, to show triangles are congruent we look at parts of the triangle, just as we can look at shoe size, pants size and height of 2 people to determine if they are twins (see photo below).

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Below is a video of a lesson I recorded on function notation using the Explain Everything app. The lesson starts by addressing the concept of function notation by connecting it to the use of the notation “Dr.” as in Dr. Nick of Simpson’s fame. The lesson builds on prior knowledge throughout with a focus on color coding and multiple representations.

This videos shows an instructional approach to teaching function notation and concepts in general and video lessons can be used for students who miss class or who need differentiation.

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A student came to me with a geometry worksheet, excerpt in photo above. Extemporaneously I created cut out sides of a triangle to help make the concept of lengths of sides of a triangle more concrete.

The concept is that the shorter 2 sides must be longer than the 3rd side or you cannot get a triangle. The worksheet is very abstract and very inaccessible. (Actually there is more to this topic but I am keeping it simple to allow lay people to focus on the instructional strategy and not the “mathy” stuff.)

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Function notation is challenging for many students yet we teachers overlook the reasons for the challenges. For example, students see the parentheses and rely on the rule they were taught previously, y(5) is “y times 5”. Because this is overlooked by teachers we often skim over the concept of notation and delve into the steps. This document is a means of presenting the concept and the use of function notation in a meaningful way. Feel free to use or revise and use the document as you wish.

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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.

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