Saw the following price tags, shown in the two photos, at an office supply store. $4 for 4 batteries or $9 for 8 batteries. To compare we can double the smaller pack to see that 2 packs would cost $8 for 8 batteries for a better deal.
Another method is to use unit rates. Rates are a measure of one quantity, with units, per 1 unit of another quantity, e.g. you make $10 per hour. To compute
Below is an example of instruction for unit rates to help a student conceptually understand (pretend that gas price shown on this pump is $2 per gallon). Say you pumped 3 gallons and it cost $6. Show the 3 1-gallon gas cans together and the 6 $1 bills together. Separate them to you have equal groups to get $ per 1 gallon. You can use actual gas cans (unused) or cutouts from Google Images.
A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.
The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”
The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”
What remains is 75% or $60. This is the “new price” which is called the sales price.
Problems like the addition problem below are often viewed by adults as straight forward. This perception can make it difficult for adults, including teachers and even special education teachers to help students who struggle with it.
I find that the math teacher candidates and special education teacher candidates struggle with breaking down math topics, especially “easy” ones like the one below, into simple steps. To help students who struggle with math breaking down the math topic is imperative. The analogy I use is to break the topic down into bite-sized pieces like we cut up a hot dog for a baby in a high chair.
For new teachers I use a formal task analysis approach to teach candidates how to cut up the math into bite-sized pieces. A task analysis for the problem above was an assignment given to a group of graduate level special ed candidates. As is common, they overlooked many simple little steps hidden in the problem. These steps are hidden because they are so simple or so automatic in our brains that we don’t think about them. See below for how I break this topic into several pieces or steps. For example, before even starting the addition the person doing the problem has to identify that 43 is a 2-digit number with 4 in the TENS place and 3 in the ONES place. Understanding that the problem is addition which entails pulling the numbers together to get a total (sum) is an essential and overlooked step. If a student struggles with a step the step can be addressed in isolation, as I show in another blog post.
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.
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.
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.)
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.
Finally, the student attempts to add without the scaffolding. I continue with color but then fade it.
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.
This is found in children’s toys with color used to guide use of toys.
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.
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.
Color was used for the same student to represent positive and negative numbers, first with concrete tokens then with colored numbers on paper.
More examples are shown below. Color helps a student focus on the different parts of an equation or different parts of a ruler.
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.
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.
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).
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.
Some differences are directly related to ADHD. Others are the result of secondary characteristics. In special education these are characteristics of a student that result not from the disability but from how the disability plays out in an academic and other settings. For example, a student with a speech impediment may be very timid and anxious in situation in which he or she may need to speak.
In math a major secondary characteristic is math anxiety. This is a performance issue vs an ability issue and it must be addressed as a legitimate obstacle for the student. I work with graduate students who still suffer math anxiety years later.
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.
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.
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.
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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.
I think use the term “more than” by pointing to the plate with more and explain “this plate has morethan this other plate.” I go on to use the quantities.
Finally, I introduce the symbol to represent this situation.
Below is the example my 3rd grade son used to explain less than to a classmate with autism. This method worked for the classmate!