Every student knows what it feels like to be puzzled by a math problem.

Sometimes the confusion happens because the teacher didn’t present a particular kind of problem in the instructional examples. Other times the student lacks the prerequisite skills needed to solve the problem. When students get stuck, some choose to dig in and practice trial and error. Others simply guess, or they stop working and take an incomplete for the assignment.

Every teacher understands the difficult choices necessitated by a finite amount of instructional time.

There are always trade-offs. In a classroom full of unique learners, should you prioritize the needs of the students with the most significant academic gaps and re-teach skills from two grade levels ago? A more balanced approach might be to teach the objective to the “middle.” Some would suggest it’s wisest to focus your attention on the higher-achieving students, and the rest will rise to meet your expectations. But what about those whose first language is not English, and the others with behavioral needs?

There are so many things to think about when planning a new lesson! ## Offer Multiple Scaffolded Examples

One of the most helpful ways for math teachers to focus their planning time is to carefully think through the instructional examples they will use. Example selection is strongly emphasized in the literature of effective instruction. A best practice for teaching math is to select multiple examples that are strategically scaffolded to cover a range of problem types. The overarching idea here is to familiarize students with many different variations of a particular concept while making explicit connections for them about the common features that tie these variations together. Research shows that students with learning disabilities or difficulty learning math benefit greatly from explicit instruction.

A series of examples can move from concrete to abstract, easy to hard, or simple to complex. The goal is to start with an example that is accessible to all students. Some concepts, such as fractions and algebraic equations, lend themselves more readily to using manipulatives and/or pictorial representations before moving to more abstract examples. Using scaffolds to sequence examples is especially important when students are learning new procedural skills (compared to when students are reviewing for a test, for example). When they see how to solve problems that range from simple to complex, students are better able to transfer their learned skills to novel situations.

Applying known skills to novel situations tends to be especially difficult for those with learning disabilities. Providing multiple examples is one excellent way to explicitly teach students how to do this.

Let’s take a look at how to scaffold instructional examples when teaching how to solve equations. ## Scaffolding Algebra Examples

Solving linear equations is a critically important standard at the heart of algebra. The ability to fluently solve equations is a required building block for much of the work in advanced mathematics. It’s also a huge skillset to master. Solving equations is the compilation of dozens of smaller objectives that build upon one another. Students need to learn to recognize the unique nature of each of these types of equations and apply the appropriate rules to solve them:

• Linear equations with one variable
x + 3 = 9
• Linear equations with a variable on both sides
4x + 2 = 8x
• Linear equations with parentheses
7(x – 3) = 42
• Linear equations with fractions and decimals
½x + 2 = -4
• Linear equations with zero, one, or multiple solutions
-7x + 3 = 2x + 2 – 9x
• Linear equations with unknown coefficients
a(8 – x) = bx – 2

If you are beginning to plan your examples for a lesson in your linear equations unit, remember that instructional examples should be scaffolded within a single objective. Do not try to “scaffold up” to show examples of future objectives as an extension or enrichment activity. Students will need explicit instruction on that future objective, too. If some of them are ready for enrichment, it is better to offer an extension problem within the day’s objective. ## Supporting Students with Significant Disabilities

Some students with moderate to severe intellectual disabilities or significant learning disabilities will need a “scaffolded down,” or modified, objective to make the learning accessible. In this case, teachers might focus on the first tier of this skill—solving linear equations with one variable. To build up to this objective, teachers might scaffold instructional examples by teaching or re-teaching these skills:

• Single-digit subtraction
• Identifying operation symbols
• Identifying variables and constants

Students will benefit from math worksheets that are visually easy—lots of white space with very little text and only a few problems per page. Use a highlighter or colored font to help them distinguish the variable from the constants, and provide boxes or lines to guide students on where to write. Offer whatever tools the students may need to accurately solve the addition or subtraction—calculators or counting chips may work well. Present the same type of problem over and over to reinforce the new skill. ## Working with Different Groups of Students

Here’s another way to scaffold instructional examples. In this problem set, the teacher is introducing how to solve linear equations with parentheses:

### Warm-Up or Quick Review

Students may practice a few problems with the prerequisite skills for the day’s objective (distributive property and solving equations with all four operations).
7(x – 3)
8(4x + 2)
x – 3 = 8

### Example 1

Solving linear equations with parentheses.

• Positive integers only
• One variable; on the left side of the equation
• No combining like terms

7(x – 3) = 42

### Example 2

Solving linear equations with parentheses.

• Positive integers only
• One variable
• No combining like terms

4(8 – x) = 2x

### Example 3

Solving linear equations with parentheses.

• Combine like terms

5x – 3x + 2(x – 1) = 7

### Example 4 (Optional—This Is Scaffolded Up)

Solving linear equations with parentheses.

• Negative integers

-5(x – 10) = 35

### Example 5 (Optional—This Is Scaffolded Up)

Solving linear equations with parentheses.

• Fraction or decimal

½x + 2 = -4

If you are a special education teacher who is co-teaching in a classroom, you may pull a group of students for small-group intervention work after the second example to continue to practice the objective at the initial level of complexity. You could also work with some of them to re-teach how to combine like terms based on data from a recent formative assessment. A quick review of combining like terms (or multiplying integers or applying the distributive property) will set students up for success with the array of scaffolded problems in this algebra example.

When teachers offer a wide range of example problems, students are much more likely to successfully apply their new skills to a broad set of problem types and real-life situations. From building foundational skills to presenting more advanced concepts, you’ll find support and guides that help you focus on concept mastery.