This link is for a drop box that contains the handouts for this presentation. Please email me with follow up questions **ctspedmathdude at gmail.com**.

This link is for a drop box that contains the handouts for this presentation. Please email me with follow up questions **ctspedmathdude at gmail.com**.

SBAC and PARC problems used to test CCSS are challenging and often draw upon context unfamiliar to students. This means students must navigate the content, problem solving and deciphering context. Below is an SBAC problem dealing with photo albums…PHOTO ALBUMS. Do kids today understand this? In the subsequent pictures you will see the work of one of my students on handouts I created that develop an understanding of the SBAC problem – note the “x-2″ at the end. The idea is to shape their ability to do such problems.

Dragon Box (link to website here) presents solving equations, proportions (and fractions and expressions) using an alternative representation and in a highly engaging game format (different platforms!).

In the photo above, the goal is to get the treasure chest by itself. To do this, the fly looking thing and the snake have to be eliminated. First eliminate the snake with the black background (a night card) by placing the other snake card on top of it (a day card). The day card snake must be placed on the right side as well. The day and night card (representing positive and negative numbers) become a circulating hurricane looking card (which represents a zero).

I used it with my 5 year old son (video here) and he could solve 2 step equations (on the game) independently within a couple hours. This is a game changer in teaching kids algebra.

This chart is what we use to teach students to count total value for given coins. Students start by placing coins in order from most value to least. We start with pennies only, then nickels, dimes then combinations. Students learn value of coins with initial matching then through working with the coins as they count total value.

At my school we are looking to develop a track for math for students with disability for whom the traditional math courses are inappropriate. Many of the topics and concepts from the traditional courses are addressed but in a practical approach with the principles of UDL incorporated. This is not a typical dumping ground situation that many of us have encountered. These are courses for students who need support in being independent and who will not benefit from being exposed to simplifying (5x + 2y – 3xy) – (7x – 8y + 9xy). Contact me if you would like more info!

Photo below shows a grid I use to identify the daily activities for my 12 high school Consumer Math students (all with IEPs) in an 86 minute block.

Our daily agenda is comprised of a warm up puzzle (to get them settled and to allow me to coordinate with the other adults – between 2 and 4) and two rounds of activities. Each column is used for a student. The “C” indicates computer use (we use IXL Math mostly and have 4 computers in the room) and the colors indicate the adult supervising.

Each student has a manilla folder which contains all handouts needed, including the daily puzzle, and the IXL modules to complete taped to the inside front cover.

In a given day we may have different students counting nickels and pennies, identifying coins, identifying bills needed to pay a given price, computing sales tax and total price and creating a monthly budget on Power Point, with photos. The system allows me to track the many details.

The following shows steps to introducing the concept of the value of money and of adding coins.

The concept of a dime is presented as 10 pennies (see below). The dime is compared to a penny, nickel and quarter using these representations. Repeated use of these representations leads into an intuitive understanding of the coins.

Next is determining the value of multiple coins. The place to start is with pennies, which is relatively easy as the number of pennies represents the value. The next step is to count dimes because counting by 10s is relatively easier than counting by 5s or 25s.

Dr. Russell Gersten is a guru in special ed. At a presentation at the 2013 national Council for Exceptional Children he explained that number sense is best developed using the number line. With this in mind I created a CRA approach using the number line.

First, the student lines up the dimes on the number line (see photo below) then skip counts to determine the cardinal value, which is the value of the coins.

Upon demonstrating mastery of counting dimes, the student moves from using coins (concrete) to a representation – see photo below.

This approach is used for nickels and then a combination of nickels and dimes (corresponding blog post forthcoming).

The link below is to a drop box that has the handouts presented in the presentation, including a PDF of the presentation itself.

Slope is one of the most important topics covered in high school algebra yet it is one of the least understood concepts. I have two observations about this. First, slope is often introduced with the formula and not as a rate of change. Second, students intuitively understand slope as rate of change conceptually when presented in a relevant, real life context. The challenge is compounded when slope is presented with the y-intercept.

In the photo I present slope and y-intercept in a context students can understand (money is their most intuitive prior knowledge). The highlighting makes it easier for them to see the context, specifically the variables. I have the students work on this handout and I circulate and ask questions.

Here’s a typical exchange – working through problems 11, 12:

- Me: “Look at the table, what’s changing?”
- Student: “the cost”
- Me: “How much is it changing?”
- Student: “20”
- Me: “20 what?”
- Student: “20 cost”
- Me: “What are you counting when you talk about cost?”
- Student: “money…dollars”
- Me: “So the price is going up 20 what?”
- Student: “Dollars”
- Me: Show me this on the yellow” (student knows from before to write +$20)
- Me: “What else is changing?”
- Student: “People”
- Me: “By how much”
- Student: “1 people…person”
- Me: “write that on the green”
- Me: “Now do this same thing on the graph. Where do you start?” (they put their pencil on the y-intercept
- Me: “What do you do next?” (they typically know to move over and up)
- Me: “Use green to highlight the over” (they highlight)
- Me: “How much did you go over?”
- Student: “1…1 person”
- Me: “Now what?” (Student goes up.)
- Me: “Highlight that in yellow.” (They highlight.)
- Me: “How much did it go up?”
- Student: “2…20…20 dollars”
- Me: “What is a rate?” (I make them look at their notes until they say something about divide or fraction or point to a rate)
- Me: “So what is the rate of change?”
- Student: “$20 and 1 person”
- Me: “Look at the problem at the top. What is the 20?”
- Student: “$20 per person.”

I point out that you can find this rate or slope in the equation, the table and in the graph.

Slope may be the most challenging concept to teach in algebra yet it is one of the most important concepts. I use the following sequence to introduce slope: rate of change, rise over run, rise over run as a rate of change. The first photo is a map of Manhattan with directions on counting city blocks. This builds on prior knowledge to introduce rise and run.

The photo below builds on the map and transitions students into coordinate planes. They are introduced to rise over run and positive and negative as indicators of the direction of a line.

The photo below combines rise and run with rate of change. The hourly wage is prior knowledge they can much more easily comprehend. A major issue is getting students to include units and to understand what units are. This would have been addressed in the previous unit on rates and proportions.

This is the handout used with a train activity in which I use battery operated trains and time them as they travel 200″. I project a stop watch on the screen as the train moves. The kids pick up right away that Percy is “slow.” As Percy is traveling I ask them how they know it is slow and get answers like “it takes a long time.” This is a concrete representation which they can draw upon as they work with the graph and calculations.

The photo below shows a scaffolded version of a Smarter Balance (Common Core assessment) test question. The original question simply shows the graph and asks for average rate of change from 0 to 20 years. Even with the scaffolding many problem areas appear: units vs variable (student wrote “value” as opposed to $), including $ with the 1000, finding unit rate, and even identifying the part of the graph at 0 years.

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