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CTSPEDMATHDUDE
Making math accessible for all students
Money in Authentic Settings
Working independently and effectively with money is a crucial component to independent living. When I started working on math for students receiving special education I was taken aback by the number of high school students who could not work with money effectively, including counting out the total value for a given set of coins.
One of the first situations I encountered involved an upperclassman who, as reported by the parent, was learning to count money by completing handouts at school. This is NOT the way to learn to handle money. Worksheets can be used to target a specific individual skill but to learn to handle money the student has to actually handle real money.
This can take the form of baby steps – learn to crawl before walking. If a student has limited money skills here is one way to get started.
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Have the student simply hand money or a card to a clerk (see photo below). This can be done while you are shopping and the student only hands over money and receives the money.
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Pick a single item that costs a couple of dollars (and some change). Hand the student an appropriate number of bills (no change yet). Have the student count out the bills for a total and hand it over to the cashier. If necessary, have the student count out the money at a table or empty aisle in the grocery store then take the money over to pay. Then the student receives the coins and hands them back to you.
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Same scenario but this time provide the student the bills and count out the pennies needed to pay. Choose an item that costs just a few cents, e.g. $2. 08. The student practices counting out bills and pennies.
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Continue this with just dimes then dimes and pennies etc.
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At some point you will want to address the concept of change returned by the cashier. To do this have the student pay with a higher bill (5 or 10 dollar bill), receive the change then count out the change at the table. Compare to what is on the receipt (see photo below).
For many of our kiddos this process can take a long time because the simple steps like counting out dollar bills takes practice. For example, students often count out money by laying the bills side by side and this takes time. This is not an effective approach to use while standing at the aisle facing the cashier.
Superhero Math
I was recruited to help a middle school student who is having a very rough time at this time in his life. It was shared with me that he likes Marvel superheroes and he is struggling with counting money and multiplication. Below are some ideas I presented for a test run and photos of the items I ordered for these suggested activities.
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For multiplication
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Put the heroes (or villains) in groups of 2 and have him count out 4 groups and compute. Use different groups and number per group. (IGNORE the numbers on the cards)
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Get a group of 10 villain cards. Pretend heroes have to travel in groups of 2 and ask how many groups to get 10 heroes to fight the 10 villains. (IGNORE the numbers on the cards). Variations of this.
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After gets the idea of groupings, focus on the number on the cards and show him two 5s and have him compute. Variations of this.
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Play a game where he draws two cards and has to multiply the cards (start with very low numbers or maybe show him a 2 card and he has to pick another card to multiply by 2.
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For Money
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Tell him he earn money to buy these figures, one at a time – a monetary version of a token economy. Have him rank them by his favorite to least favorite and come up with a price for each with his favorite figures costing more. Start with the least favorite and make the price such that with a little practice he could count out the coins to pay for it. Maybe 17 cents with dimes and pennies. He has to count out the money correctly and independently to actually buy the item.
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Other options
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If he needs work with addition you can play WAR in which 3 cards are played and each person adds to find the total. For subtraction do the same with 2 cards.
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You can play subtraction in which one person has superheroes and the other has villains. In order for a villain to win a villain card has to be higher than a hero card by 3 or more.
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You can write an 11, 12 and 13 on the J, Q, K cards respectively. All the games can be presented though Direct Instruction – I do, we do, you do. The You do can be used as daily progress monitoring. If he needs prompting this can be recorded. This can be used for your progress reports. Attached is a data sheet I use for activities.
Symmetry Made Accessible
The work shown below posted on LinkedIn by Maria Priovolou. I think this is awesome.
The photo below shows a focus on just the vertical axis and the student has to reflect one object at a time. This is a nice task analysis approach. The stamp creates the objects which makes it hands on and a little different from just mathy work.
This hands on work can be followed with work on this website. In the photo at the bottom you see an example problem. This can make reflection more concrete and eventually more intuitive for the student.
Making Proportions Meaningful (and Therefore Accessible)
A student reported to our schools math lab where I reside. He had a handout on proportions shown in the photo below and stated that he didn’t know what to do.
I find that in the vast majority of situations like this the student lacks the conceptual understanding of the topic. As is typically the case, I started my sessions with the student by focusing on something he more intuitively understood. Teens know money, phones, games, music and food.
In this case I started by showing him a photo on my phone shrunk then enlarged the photo and talked about how I could double the size of the photo. We talk about what doubling means then I show him a handout with the photo in two sizes (below).
I explained that the small photo was 3×2 inches and that I wanted to enlarge it. The bottom of the big photo is 6″ but I needed to figure out the height (vertical length) which is marked with an X.
I had him figure out the height (4). Then I explained that proportional means the shape is the same but bigger or smaller. In this case both the side and bottom were multiplied by 2. Then I showed him the “mathy” way of doing the problems. This progressed towards the handout he brought into math lab. By the end he was doing the proportions independently.
Velocity, Acceleration, Speeding Up and Slowing Down
DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.
We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.
The velocity can be graphed (the red curve below). Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left. The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)
Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.
As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.
The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!
It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.
Lego Fractions
The photo below is courtesy of Robert Yu, Head of Lego Education China, as shared by Jonathan Rochelle, Director for Project Management at Google.
The use of Legos shown here is a classic (and wicked clever) example of manipulatives.
Before writing the actual fractions students can use drawings as shown below. The sequence of manipulatives, drawings then the actual “mathy” stuff constitutes a Concrete-Representational-Abstract (CRA) model. Concrete = manipulative, Representational = picture, abstract = symbolic or the “mathy” stuff.
Comparison Shopping – Real Life Example
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
$4/4 batteries = $1/1 battery vs $9/8 batteries = $1.13/1 battery
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.
