Elapsed Time Scaffolded

The photo above shows a scaffolded handout to break down elapsed time for a student. The problem is divided into 3 parts: time from 10:50 to 11:00, time from 3:15, time from 11:00 to 3:00 (see photo below). This is based on how we may compute elapsed time by focusing on minutes then on hours. Notice the 3 clocks (in photo above) with no hands which allows the student to engage the clocks by having to determine and show how many minutes passed, e.g. 10:50 to 11:00.

The final answer would be 4 hours and 25 minutes.

One More or One Less Scaffolding

The photo above shows a screen for a hundreds table was shared by one of my students in a Math for Children graduate course. She found it on Pinterest for use a class presentation. I love this idea and came up with some revisions I think can make it more effective. It seems to me that this screen may be too busy with 4 different numbers showing. Additionally, the view of the other numbers outside the screen could be distracting.

Below are a couple revisions I would suggest.

Making Slope Less Complicated

Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

• The yellow numbers represent time
• The orange numbers represent altitude
• The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.

Introduction to Slope

Slope is one of the the most important topics in algebra and is often understood by students at a superficial level. I suggest introducing slope first by drawing upon prior knowledge and making the concept relevant (see photo above).  This includes presenting the topic using multiple representations: the original real life situation, rates (see photo above) and tables, visuals,  and hands on cutouts (see photos below).

A key aspect of slope is that it represents a relationship between 2 variables. Color coding (red for hours, green for pay) can be used to highlight the 2 variables and how they interact –  see photo above and below.

The photo below can be used either in initial instruction, especially for co-taught classes, or as an intervention for students who needs a more concrete representation of a rate (CRA). The clocks (representing hours) and bills can be left in the table for or cut out.

Analyzing a Graph

Students can hit a road block at the steps that appear to be very simple. For example, in the problem below the students are prompted to find the highest point on the graph. Many think the graph refers to the entire coordinate plane and they pick 5 as the high point. It is the highest point on the y-axis but not the graph. I introduce the problem by highlighting the actual graph in pink and explain that this highlighted line is what is meant by the graph.

The use of color also helps students distinguish between the x and y axes and what the variables x and y represent in the context of the problem (# minutes and # kilometers in this problem) – see photo above. This problem also involves plugging in a # for x (blue) IN the function (red). In the photo below you see how I use color to help emphasize this.

Modified Multiplication Table – Area Model Included

The idea is that the student will have to count squares. By doing so the student is more engaged (or less passive) in determining the product and has to engage the visual representation.

Here is a link to the document.

Fraction Word Problems Scaffolded

Above is a handout used to scaffold student work with fraction word problems (more on this below).

Here is a type of word problem I recently encountered in working with a student with special needs: There are 60 students. 3/5 of the students are girls. How many students are girls? The student struggled with this problem in regards to the concept of fractions and in determining a fractional amount of a total.

To address the concept of fractions I used the handout seen in the photo above. (The handout can be printed in color to show the actual colors of the birds – see this handout.) A pink highlighter (red is too dark) is used to help the student connect the actual red birds with the number of redbirds used in writing the fraction.

To work out the 60 students problem the following approach is used to develop conceptual understanding – see photo below. (Note: for students in upper grades who struggle teachers often turn first to showing how to solve by multiplying. Students who are working at below grade level typically need the conceptual piece to be addressed first.)

1. Break the all the items into groups based on the denominator, in this case it is groups of 5.
2. Mark the fractional amount in each group, in this case mark 3 of the 5 circles.
3. Find the total number of circles marked in.

To scaffold this approach I use a task analysis approach and break the strategy into steps.  First I use a handout focusing students on circling a fractional amount in each group – see photo below.

The next step is to count the number of items circled (or marked).

The next step is to use a situation where the items to circle are not colored (the circles in the student word problem are not colored but are hand drawn figures). In the photo below Students are tasked with circling and counting.

Then students take the next step is to answer the question to find the total number.

Finally, students are tasked with creating their own drawings before circling and counting.

UDL Approach to Presenting Notes

The photo above shows an excerpt from the presentation of notes in an algebra class using an UDL approach. The following strategies are implemented:

• Graphic organizer
• Color coding (notice that the slope, which is a rate of change, is green for go – movement and notice the consistent use of the colors for prior and new knowledge)
• Connections to prior knowledge
• Chunking (before attempting numbers the presentation focuses on the contrast between new and prior knowledge)

This allows for Multiple Means of Representation as found in the UDL Guidelines.

Assessment of a Math Objective

List all the steps for the objective. Use this table (above) as a pretest to identify gaps.

Provide instruction on the gaps. In the photo below I used color coding to show what to multiply and scaffolding to align the digits in ONES and TENS place. NOTE: I provide the problems with some steps already completed to focus on the steps for which gaps were identified.

After providing instruction on the steps with gaps data is collected on mastery of these isolated steps. NOTE: The problems are identical in nature to the gaps and the problems used in instruction. (Link to the data sheets – WORD so you can revise.)