Multiplication by 2-Digit Factor – Scaffolded and Color-Coded

This was used for a student with ADHD who understood how to do the multiplication but would rush and repeatedly made simple mistakes. This grid and color-coding strategy was used as a means of slowing him down. He had to alternate between highlighting and writing each product for an individual multiplication of two digits. This turned out to be an effective way to teach multiplication by 2-digit factors, in general. Here is how this works.

First highlight the ones-digit in and the row for the product that results from the ones-digit. This helps unpack the place value and why the algorithm works. Note: use a lighter color of highlighter (you will see why).

Highlight the ones digit in the top factor. Multiply the ones digits. Write the product. This is where the student alternates, which can allow for thinking through the steps.

Continue highlighting and writing products using the ones digit from the bottom factor.

Now use a darker highlighter to highlight the tens digit in the bottom factor, as well as the tens row at the bottom. Because the 3 is in the tens place, we write a zero. This unpacks the place value.

As was done with the yellow highlighter, alternate between highlighting digits to multiply and write the product in the row below. The darker highlighter is used second to make it visible when drawing over the previously used lighter color.

For carrying (regrouping), the top row can be split and the color can be used for the digits that are carried.

Here is a link to the handout used for these photos. It contains the two problems shown in this post along with blank templates. Here is a link to another post that shows a scaffold I use to unpack the carrying of a digit in multiplication.

Time on an Analog Clock – a Chunked Approach

Telling time on an analog clock is challenging for many students, especially some with special needs. I worked with a middle school student with a disability one summer and after a few lessons he scored 100% over two days on telling time. Below shows the progression I used with him. I used a task analysis approach of breaking the task into smaller steps and chunking the steps to introduce an additional task demand incrementally.

I use math-aids.com worksheets for time and for many topics because it provides dynamic worksheets in which users can choose features. This helps to enable implementation of a task analysis and chunking approach.

The first chunk is whole hour time, which is an option on math-aids. The clocks produced by have color coded hands, with green for hours and red for minutes. I use use additional color coding through highlighters because the handouts are likely printed in B&W and because it engages students kinesthetically.

The second chunk is time with minutes under 5 minutes. Math-aids allows a user to choose specific times and will create clocks with those times. To provide a visual aid, students can write out the numbers on the handout or they can be printed or handwritten on the master copy. You can snip out the clocks shown, paste into a WORD document and then add the numbers. Note: I do not jump to half hours and quarter hours until last. I want students to focus on hours and minutes. This is analogous to counting money. I don’t introduce cents after talking about half a dollar.

The next chunk is time with minutes between 6 and 10 minutes. This is how I introduce the 5 minute mark and have them count on from 5 (see the 5 in the middle bottom). This leads to all the tic marks for 5s.

The 5s are the entry point to navigating the entire clock. I introduce the tic marks for 5s without the numbers for the hours. Students can be prompted to draw in red minute hands for a given numbers of minutes in 5s. If you want to make a handout for this, save the image below and crop.

The students are then given the minute hands and are prompted to identify the minutes as a multiple of 5s (you don’t say “multiples” but can say “in 5s”). An option is to have them highlight the minute hands at first then fade the highlighting. They do this first with no numbers for hours then with numbers for hours – an option on math-aids.

Then students are asked to tell time by identifying the fives and then counting on (as they did with time with minutes under 10). Here are some options.

  • You can have them highlight.
  • They can focus on just the 5s and write the multiple of 5 preceding the minute hand.
  • They can count on from the 5s multiple they wrote, e.g., “15, 16, 17”

Focus first on minutes below 15 as the hour hand is close to the hour (top row in image below). Then address time with minutes between 16 and 30 because the hour hand has moved further away from the hour and it starts to get tricky for students to determine the hour. To address this, you can shade in the tic marks between the hours. Notice that I do not shade the subsequent hour. This also sets the stage for time when the hour hand is close to the subsequent hour (next chunk).

Time with the minute hand on the left side is tricky because the hour hand is close to the subsequent hour. The aforementioned shading can help. I also find it useful to have the student notice that the hour hand is not at the 12 yet, but almost. You can have students draw the marking and word as I did below.

Finally, there is a need to generalize. You can print images of clocks from a Google Images search into a handout and use the same strategies from above. This would be followed by actual clocks.

Piecewise Functions – Introduction

Here are images from a handout that serves as an introduction to piecewise functions. The focus is to develop conceptual understanding of piecewise before attempting to graph independently. The work is divided into chunks to reduce the level of task demand at a given time in the process.

The first section has an application and introduces the idea of pieces addressed together. The graph is discrete which allows the students to see the points in lieu of looking at lines.

Pages 2 and 3 provide scaffolding for graphing. Page 2 presents the sections separately. Page 3 pulls them together, but with the intervals physically separated into columns over the graph.

Finally, the students are introduced to the function notation, with additional scaffolding. They are also asked to identify y-values for given x-values in function notation to help connect x-values with different pieces.

Modified Multiplication Table – Area Model Included

The idea is that the student will have to count squares and eventually skip count by how many rows. By doing so the student is more engaged (or less passive) in determining the product byy engaging the visual representation. I am interested in feedback and will revise if this could be useful.

Here is a link to the document.

Combining Like Terms – From Concrete to Symbolic Form

Here is a Jamboard (see photo at very bottom for how to make a copy to edit) that presents like terms as visual manipulatives and then eases into the symbolic form – the “mathy” stuff. The following shows each slide as is and how it looks after completion.

Start with two groups of common items, pull them together because they are alike, and compute the total. This slide is designed to introduce the students to how the manipulatives work for this Jamboard and to introduce them to the concept of like terms.

This slide introduces actual like terms. It is the “mathy” representation of the previous slide and is the most basic form of simplifying algebraic expressions. (The phrase, simplifying algebraic expressions, can be introduced later to allow the focus to remain on “like terms.”)

This slide introduces different types of terms at a conceptual level and how they are rearranged into like groups. It also introduces the use of a binary operation between the two groups of terms (vs adding everything together as the students are wont to do). This slide also establishes color coding by like terms, which is useful for when the work shifts from concrete form (manipulatives) to abstract form (written symbols).

This is the mathy version of grouping by like terms. This slide is crucial as it presents the Associate Property with the binary operation symbols acting for a moment as unary operations. In other words, the addition symbol follows the 2b, the 4t, and the 1b before latching on to another term. It is easier to address this while all operations are addition. Present the original problem as an algebraic expression and not individual pieces “3t plus 2b plus 4t plus 1b”. NOTE: I keep the coefficient of 1 to reduce task demand – one less thing to think about. I address the implied coefficient of 1 after students have had ample exposure to like terms.

The use of 1 dollar bills is intended to introduce constants. I fluctuate between whether to write the terms for the bills with or without units (4 dollars vs 4).

At this point, I suggest giving students independent practice with expressions that have addition only before moving on to the following slide.

This slides shows how I introduce subtraction and negative terms. The image shows a woman eating a taco, hence it cancels one of the tacos.

I present the eating 1 taco image as a negative, with the terms separated as opposed to being an expression. This allows the students to see the “-” symbol as a unary operations (negative) and then as a binary operations (subtraction). In other words, they see the symbol “attached” to the term. It is a prelude to the use of the Associative Property with subtraction.

The new image is of a person eating a burrito. The slide introduces the concept of the negative term in the Associate Property.

**The students are now presented with the what is likely the most challenging aspect combining like terms, which is the “-” fluctuating between being a unary and binary operation. The original problem is again presented as an algebraic expression, “2b minus 2t plus 4t minus 1b”. The minus 2t is converted into negative 2t while the minus 1b is converted temporarily converted into negative 1b as it is moved, but then is converted back into minus 1b. This is the opportunity to unpack this situation. You redo the problem with the 4t minus 2t and negative 1b + 2b (and add the plus symbol) as part of the discussion. You can also duplicate this slide and revise into different problem.**

The next two slides have the color and the “?” sticky note faded. They can duplicate an existing sticky note to record the final answer (or you can add in the “?”).

Finally, students are presented static algebraic expressions. I return to color coding as this is a support they can take with them to handouts. Eventually, the color is faded on the handouts, but they can still use shapes to indicate the different types of terms.

Copy the Jamboard and to edit and use it.

Introduction to Pythagorean Theorem

Here is a link to the Jamboard (make a copy to edit – see photo at the very bottom). The directions are on each slide. The photos below show what the results should look like.

To make a copy, click 3 dots and choose copy.

Plotting Points Introduction

Plotting points is challenging for some students, even in high school. This approach uses the analogy of setting up a ladder for the x-value and climbing the ladder for the y-value.

Jamboard with Ladder

This activity is conducted on a Google Jamboard with moveable objects (see photo at the bottom on how to access it).The activity draws on prior knowledge, which allows for the steps for positioning the point point to be meaningful. The x-axis is established as the ground, with an image of green grass superimposed over the axis. The ladder is climbed, moving toward the sun. This aligns the ordered pair with the axes. It also allows for color coding on coordinate planes on subsequently used handouts.

The steps are listed below and can be viewed on Youtube or a FB Reel showing the steps.

  • Move the ladder to the appropriate location on the ground.
  • Position the dot at the bottom of the ladder on the x-value.
  • Move the dot up the ladder to the appropriate y-value.
  • Pull the ladder away.

Jamboard without Ladder

Next, the ladder is faded but the color is maintained. 3 is green so move along the grass to the 3. Then yellow 5 so move up 5, towards the sun.

Finally, the colored numbers are retained but the grass and sun are faded. References to the green grass and sun can be used as necessary, even with highlighters on the Jamboard.

Handout

The Jamboard can transition to a handout with matching coordinate planes. The grass and sun are faded. If a student is stuck, you can return to the Jamboard to model the problem.

Students can use highlighters as necessary to replicate the grass and sun numbers. The highlighters can be faded to result in a regular plotting a point problem.

Accessing Jamboard

To access the Jamboard, make a copy.

Information Processing Analogy – Big Picture

Effective instruction is effective because it addresses the key elements of how the brain processes information. I share an analogy to help adults (parents and educators) fully appreciate this.

Information Processing Model

Below is a model of information processing first introduced to me in a master’s course at UCONN.

Here is a summary of what is shown in the model.

  1. Human senses are bombarded by external stimuli: smells, images, sounds, textures and flavors.
  2. We have a filter that allows only some of these stimuli in. We focus on the ones that are most interesting or relevant to us.
  3. Our working memory works to make sense of the stimuli and to package it for storage. It is like a computer, if there is too much going on, working memory will buffer.
  4. The information will be stored in long term memory.
    • Either it will be dropped off in some random location and our brain will forget the location (like losing our keys)
    • Or it will be stored in a file cabinet in a drawer with other information just like it. This information is easier to find.

Analogy to Classroom Learning

Here is an analogy to what happens during school instruction. You are driving down the street, like the one shown below.

There is a lot of visual stimuli. The priority is for you to pay attention to the arrows for the lanes, the red light and the cars in front of you. You have to process your intended direction and choose the lane.

Other present stimuli may be filtered out because it is not pertinent to your task: a car parked off to the right, the herbie curbies (trash bins), the little white arrows at the bottom of the photo. There is extraneous info you may allow to pass through your filter because it catches your eye: the ladder on the right or the cloud formation in the middle.

Maybe you are anxious because you are running late or had a bad experience that you are mulling over. This is using up band width in your working memory. Maybe you are a relatively new driver and simple driving tasks eat up the bandwidth as well.

Impact on Students

For students with a disability that impacts processing or attention, the task demands described above are even more challenging. A student with ADHD has a filter that is less effective. One with autism (a rule follower type) may not understand social settings such as a driver that will run a red light that just turned red. Another with visual processing issues may struggle with picking out the turn arrows. Their brain may start to buffer, like a computer.

What is Buffering? — Causes and How to Stop It - Dignited

Specific Disabilities

Effective instruction would address these challenges proactively. Here is a video regarding learning disabilities (LD) that summarizes the need in general for teachers to be highly responsive to student needs. This link is for a great video that helps makes sense of what autism in terms of how stimuli can be received by those with autism (look for the street scene). Another is a video of a researcher explaining how ADHD responds to sensory input (he gets to a scenario that shows how impulsiveness can be a factor).

What to Do

To address these challenges:

  • Reduce the amount of information presented in a lesson segment, i.e., chunk the lesson.
  • Use color, e.g. highlighters – this helps students see the different parts of a problem
  • Use hands on and visual representations in lieu of words – words are symbolic and abstract, start with forms of information easier to process.
  • Connect information to prior knowledge or make it relevant.
  • Scaffold the work to provide supports for unpacking the concept, following the steps, or identifying the parts.
  • Relevant situations – learn by doing. Have the instructional setting mirror the real life setting as much as possible. Better yet, conduct instruction in the real life setting.

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

Color – Easy to Implement Strategy

When my son was in preschool I asked him who was in his class. He replied, ” Natalie, she’s the yellow heart.” Children learn color before they learn words because it is easier to process.

preschool shapes and colors

This is found in children’s toys with color used to guide use of toys.

keyboard for childrens book

The obvious use of color in real life in traffic lights. The colors represent different concepts with red being used universally in the U.S. as representing stop. Color is used to partition an object into sections, as often seen in maps of areas with different sections. Think of how many highlighters are sold to college students to help them highlight key passages in textbooks.

traffic lightsFenway Park, Boston Red Sox's Ballpark - Ballparks Of Baseball in Miller Park Seating Chart With Seat Numbers Image

The use of color help convey information, especially sections of a whole is an effective and easy to use instructional or support strategy.

The top two images below show my earliest attempts to use color. The student for whom this was used was a 7th grade student with asperger’s who tested in math and reading at a 1st grade level.

In lieu of referring to the “horizontal line” I can refer to the “yellow line” as in “find the yellow 3” for plotting the point (3, -2). Color, as in the aforementioned yellow heart, is much more intuitive for students, especially those with a disability.

coordinate plane

Color was used for the same student to represent positive and negative numbers, first with concrete tokens then with colored numbers on paper.

adding integers chips and colored pencils (2)

More examples are shown below. Color helps a student focus on the different parts of an equation or different parts of a ruler.

linear equationsruler

Color can also help organize a room into different parts. Each color represented different courses I taught, e.g. green was used for algebra 2. The room is more organized because of the sections outlined in color. Consider how this can help a student with ADHD, autism or an executive functioning disorder.

 

classroom