This post provides details about a handout for simplifying rational monomial expressions. It incorporates a couple strategies to make the simplification of rational monomial expressions more accessible. The strategies include address prerequisites skills ahead of time, chunking, and scaffolding. This incrementally walks the students through the steps.
The Pages of the Handout
The handout has 3 pages.
Page 1 is an initiation with two parts. There is a review of prerequisite skills aligned with the new topic. There is also a preview of the new topic with scaffolding to separate the factors into individual fractions.
Page 2 provides a Before and Now to draw upon student prior knowledge of simplifying using exponents rules. This is followed by scaffolded steps to separate the expression into individual fractions for each type of term (e.g., Xs). This provides a load reduction for what the student has to focus on.
Page 3 involves negative and 0 exponents with an additional step to address each.
This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.
The slide show below presents all 4 pages.
The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.
Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.
Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.
For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).
This post presents a Google Jamboard manipulative activity to help scaffold the act of subtraction which helps unpack the concept of subtraction.
The Jamboard can be individualized with Google Images. The can allow for context. In this example, maybe the context is there are 7 players and 4 have an injury or on COVID protocol and have to sit out.
The artifact also incorporates scaffolding, color coding, and manipulatives. Subtraction is an operation, which invokes a verb. The kinesthetic aspect of the manipulatives helps to unpack the concept of subtraction.
Write the problem, using color.
Circle the starting amount in the row as the same color as the initial number.
Populate the row with the images of interest to the student.
I work with students from elementary school to college. At all levels there are many students who struggle with multiplication. When I work with a student on multiplication I focus on skip counting as opposed to multiplication facts. Skip counting connects to multiplication as repeated addition, which is the foundation for scale and proportion. To do this, I scaffolded the skip counting and connect it to prior knowledge. This post provides details for this approach, which is presented in a handout.
The first page of the handout provides an overview.
Sections of the Handout
Note: the image of the 10s shows mistakes that are not in the actual handout.
Access to the Handout
Here is a link to the document. As stated in the overview on page 1, the elements of this handout may be cropped out and used individually. At least, the scaffolded rows can be covered to prevent the students from copy from previous rows completed.
I am interested in feedback on how to make this more useful or effective.
Tenths vs Tens…Hundredths vs Hundreds. Problematic for many students. I believe this is a conceptual problem. This post provides an approach to unpack the concepts through money in a scaffolded handout.
Money is likely prior knowledge for many if not most students, and is a relevant context. This handout attempts to leverage interest or knowledge of money to unpack decimal place values. In the first page, the concept of “tenth” is addressed with dimes as 1/10 of a dollar. Similarly, “hundredth” is addressed with pennies as 1/100 of a dollar. A key point to consider is that US monetary system base unit is a dollar. More on that in the other pages.
Hundreds to Hundredths
The handout aligns each place value with the appropriate currency. This is followed by writing each number in numeric form and then word form with the place value table as a guide. To enhance the word part, you can highlight the each place value in money, digit, and word in the same color (e.g., the “2” in yellow).
Also, note the shading. The dollar as the base unit is in the center and shaded the darkest. The tens and tenths are shaded the same as they are a factor of 10 from ones. (I don’t reference the term with students.) Same for hundreds and hundredths.
My focus is on working with students with special needs. Many struggle with rote memorization, including for multiplication facts. I find that skip counting, with scaffolded support in the learning process, provides them access to multiplication and therefore division. To access division, I use an approach of skip counting to find a missing factor and then connect this to division. This post provides details of a handout using this approach.
This handout focuses on connections to prior knowledge of skip counting and finding a missing factor. The students then make an explicit connection by rewriting division problems as missing factor problems. The handout is linked at the bottom.
If students are struggling with multiplication, they are likely having trouble with skip counting. I start with a warm up on skip counting with the numbers that are easiest for students to skip count. Note: you can start with 2, 5, 10 only if necessary.
I have students solve a missing factor problem using a provided skip counting row. Then they are shown that the problem can be rewritten as a division problem which has the missing factor as the answer. That is, division is another way to write a missing factor problem. You can use factor tree handouts and have students practice rewriting the problem as a division problem. Note: I see that most worksheets are used for prime factorization. Use the first two branches as shown in the image below.
The students are then presented a math sentence only for missing factor. They are to solve for the missing factor. Then they rewrite the math sentence into a division sentence and solve again. I have a separate column to help emphasize that they are lookin to solve a division problem. They have to see the division problem in isolation and then write the quotient.
Finally, the students are presented division problems and rewrite as a missing factor problem. Their mental process can be as follows: “2 times what gives me 10?” and then they skip count by 2s until they reach 10. This can be supported with multiples rows as shown in the factor tree page. A blank page is provided. You can give students a division worksheet and have them copy the problems into the handout.
Absolute value is a challenging for many students. Absolute value equations add an extra layer of challenge. This post presents a scaffolded handout. It starts with an initiation addressing absolute value and then is followed by scaffolded steps for solving that address written and mental steps.
The scaffolding walks students through mental steps to unpack the underlying concepts. The first is a general concept of absolute value. The next two are specific to the problem. At the end, students are prompted to check the solutions to reinforce the concept of two possible values, one being negative and one positive. Note: this is not addressing the case of =0.
The scaffolded handouts are preceded by an initiation. The focus is to unpack the concepts underlying the equation steps. The key is for the students to understand that there are two possible solutions (aside from =0). The handout addresses the reason for two solutions and provides work with the absolute value symbol in this context. Here is a link to a Jamboard that helps unpack the concept of absolute value.
There are 3 practice problems with the template. I use 2 and 6 to help compare the different situations. There is another page with blank templates.
I have found simplying expressions to be one of the most challenging Algebra 1 topics. This post shows a scaffolded handout approach to simplifying.
Scaffolding Like Terms
I have attempted to provide a deeper understanding of “like terms” in this post. This handout may be a useful follow up or it may be the entry point for simplifying.
The scaffolded handout focuses attention on the problem being an expression and on unpacking what simplifying and like terms mean. This is followed by a sequence of steps to address each mental and written step.
An effective strategy is to color code, showing which terms are like terms.
Students start work on solving equations in 6th grade. They work on it in 7th and 8th and into algebra classes in high school and college. Despite this, many struggle at all levels with solving. I have witnessed and heard about this at each level. This post addresses one possible reason, which is the students are not grasping the concepts. This may be due to the way equations are presented. This post unpacks the possible reason and presents a scaffolded handout to unpack the concepts and vocabulary.
Add-on Steps We Teach
The image below shows a earlier scaffolded handout I had used for years. It includes two elements that in my experience are common: referring to the “sides” of the equation, and the vertical orientation of the step with inverses.
I think these are “add-ons” that we teachers incorporate into math. I assume there are advantages and disadvantages for each. In this case, the disadvantages as I see them are a follows.
The term side may be somewhat ambiguous. We use it in the contexts of which side are you on, stand off to the side, side of a building, there are two sides to every story etc. We know this because most of us probably have experienced something like what is shown in the image below left.
The vertical orientation (image above “-1”) does not produce an expression that they have seen before. We are changing the rules of the game and then wonder why they are confused -see another in image below right. (I have used my share of add-ons.)
Explanation of the Handout
I am attempting to wean myself off of the use of add-ons. This handout The handout scaffolds both mental steps and written steps. In addition to the skill based steps, they focus on concepts and vocabulary. Here is a YouTube video showing how it works, shown on a Jamboard accessible using this link. (You have to make a copy to access it.)
The first 5 steps focus on mental steps which are identifying components of the equations using vocabulary. The steps prompt them to write what their thinking is. The example can be presented in “I Do” fashion with think alouds.
The solving steps guide a horizontal orientation. This allows students to focus on the expressions as a whole and not in split-level fashion.
Because the expressions with steps included are horizontally oriented, it is easier for the students to see the expressions as a whole and then simplify them.
The 0 is written and addressed with an additional step to highlight the identity.
I considered adding another step for the students to indicate that the final equation shows the solution.
The second sheet has blank templates. One possibility is that you could assign students problems and they can complete the first 4 on here.
Solving systems of equations using the Substitution Method may be the most task intensive algorithm the students learn. It is easy to overlook concepts as the students attempt the numerous steps. This post provides details about a scaffolded handout that guides students through the mental and conceptual steps, as well as the traditional written steps.
The focus is on systems with a variable that has a coefficient of 1. The variable is not isolated. The entry point is typically with the variable already isolated with the written step being substitution. This handout works for both situations. For the variable already isolated, step 2 can be skipped.
The Scaffolded Handout
The scaffolding addresses the skills used the method but also reinforces vocabulary and concepts.
The first step is a mental step but the students are asked to circle as a means of focusing attention on this step. It is useful for a think aloud as the teacher talks through the process.
Substituting an algebraic expression in for a variable allows for focus on the terms variable, expression, equation and the idea that a variable is a simple expression being replaced by another one.
The solution is an ordered pair. Often, students may stop once they find the value for the first variable.
Page 2 of the handout is scaffolded to guide the students through the steps I found to be problematic. Replacing a variable with an algebraic expression is the central component of this method and is new to students at this point. previously, they would have substituted in only numeric values for variables.