Mar 01 2019
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Consumer Math

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.

  • 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.

Luisa at counter at BN

  • 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.

  • 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.

  • Continue this with just dimes then dimes and pennies etc.

  • 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).

Luisa at table with money

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.

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Feb 28 2019
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Concrete-Representation-Abstract, Connections to Prior Knowledge, Consumer Math, instructional strategies, Number Sense

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.

  • For multiplication

    • 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)

    • 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.

    • 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.

    • 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.

marvel playing cards

  • For Money

    • 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.

Avengers Figurines

  • Other options

    • 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.

    • 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.

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.

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Feb 28 2019
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Concrete-Representation-Abstract, Technology

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.

symmetry vertical axis only

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.

top marks symmetry

top marks symmetry problem.JPG

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Feb 26 2019
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Algebra 1, Connections to Prior Knowledge

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.

2019-02-26 10.38.31

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.

2019-02-26 10.37.27

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Feb 22 2019
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Connections to Prior Knowledge, Higher Level Math, instructional strategies

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.

velocity vs speed

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.)velocity graph with tangents

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!

speeding up and slowing down

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.

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Feb 21 2019
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Concrete-Representation-Abstract, instructional strategies, Number Sense

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.

Lego fraction model

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.

0214192340_2

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Feb 20 2019
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Concrete-Representation-Abstract

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

2018-11-18 15.49.442018-11-18 15.49.40

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.

gas price rate

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Feb 19 2019
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Concrete-Representation-Abstract, Scaffolding

Sales Price Entry Point

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.

IMG_20180526_165232.jpg

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Feb 17 2019
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instructional strategies, Number Sense

So Easy?!

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.

0217191819_2

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.

0217191815_2

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Feb 16 2019
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color coding, Concrete-Representation-Abstract, Connections to Prior Knowledge, Number Sense

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

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