## Helping students understand and implement a monthly budget is challenging, especially for students with disabilities that make it harder for students to conceptualize abstract ideas. I previously posted about a full budget activity. This post shows a means of scaffolding the concept of partitioning money in a budget context. The idea is to keep it simple for now and build from there.

### Set Up

A parent of a student I support came up with the following idea. We start with a couple major budget items (rent, groceries, utilities) and the temporary idea that the remaining money is discretionary (not the word we use with the student). Money is printed (legal if the printed bills are small enough and only 1 sided) in lieu of fake money that does not look like the bills they would see.

The activity is guided by slides on a Google Slides presentation (link at bottom of this post). Note: the activity can be rerun as needed and the Google Slides slides can be copied, pasted, and information removed. This allows you to keep a record of each trial with this activity.

### Job and Pay

The student can either search for a job on a site like Indeed.com or an ad for a job can be provided. The hourly rate is established and the student is prompted to compute the total pay on Google Calculator to allow a screenshot to be produced.

The student then uses the chart to provide a visual and scaffolding to compute the total pay for a month. I go with 4 weeks of 5 work days each, with no taxes to keep it simple.

The student counts out the money, first by grouping hundreds together to get a \$1,000. Then the total is moved next to the envelopes.

The pay is entered into a bank balance table to provide practice with the format of a check register. This helps provide structure and having the money counted out on the table allows the student to see a concrete representation of the bank balance table. (Note: I slide the money to the left to allow space to move the money to the envelopes as the student pays bills.)

### Paying Bills

The first bill is rent. The student is prompted to search for an apartment on a website like Apartment.com, take a screenshot, and paste into a slide.

The student then pays the bill by counting out the money and sliding the money towards the envelope.

The student then enters the rent into the bank balance. I then point to the money pile on the right in image above and refer to it as rent. I then point to the rent entry into the bank balance. Similarly, I point to the pile on the left, refer to it as the balance and count it out, then point to the new balance in the table. This provides a concrete representation for the bank balance.

I found a website that provides average bill amounts for our state. The student clicks on the link, takes a screen shot of the average costs, and pastes it into the slide.

We focused only on heat and electricity. The student identifies both amounts (I round to the nearest 5 to keep it simple) and then pays both by moving the money over.

Both bills are entered into the bank balance. I then point to the two piles of money used to pay the bills, point to the entries into the table below, point to the pile of remaining money, and point to the entry into the balance in the table below.

Finally, the student makes a shopping list of food items for all 3 meals for the week. To make it easy, we can assume the the same meal each day. The student is provided a lot of leeway in what he or she chooses and what amounts. The amounts they choose may not be enough for a week. That can be addressed in grocery shopping activities conducted in isolation.

The student shops online for the items and takes a screenshot of the cart.

The student completes the table to determine the total cost for a month.

The student moves the money over and then this total cost is entered into the bank balance. The same comparisons between money piles and cost and balance are presented. Then the remaining money is free to use for whatever the student wants. At this point, you can have the student go shopping for clothes or whatever.

Here is the Link to the Google Slides file. You can make a copy to access it.

## Graphing linear functions may be the most important topic in Algebra 1. While proportional reasoning is a prelude to functions, this is the first formally identified function presented to them. The graphing leads to slope and intercepts, beyond the entry point for graphical representations to functions. This post presents an activity that can serve as the entry point for linear functions.

### Overview

The activity is presented on a Google Jamboard, which provides manipulatives. It begins with a relevant context for students, money and being paid for a job. This allows them to engage the function using money. Before using numbers, they engage the work context through images. They are presented the table and then graph representations of the function before getting into the equation. Here is a FB Reel showing the movement of the images.

### The Slides

There are 3 categories of slides. Here is a description of each.

• Table and graph clocks for hours and dollar bill for the money.
• They graph the whole hours first, then fractional hours (1/2 and 1/4) to see that there are points “squeezed in between each other. This leads to the idea of infinite number of points. In turn, this leads to the idea of the line are a visual means to present all the points. The points can be presented as solutions. Hence, the graph presents all the solutions for the function.
• Table and graph with numbers on sticky notes that can be moved from the table to ordered pairs to positions on the coordinate plane.
• The equation, with sticky notes to show numbers substituted in for the variables and then moved to ordered pairs with parentheses.

Here is a link to the Jamboard. You need to make a copy to access it.

## This post presents a Google Jamboard manipulative activity to help scaffold the act of subtraction which helps unpack the concept of subtraction.

### Overview

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.

### Steps

1. Write the problem, using color.
2. Circle the starting amount in the row as the same color as the initial number.
3. Populate the row with the images of interest to the student.
4. Physically take away the identified amount.
5. Write the remaining amount as the answer.

Here is the link to the Jamboard. You need to make a copy to access it.

## 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.

### Overview

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.

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.

### Overview

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.

### Thousandths

here is a link to the handout.

## 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.

### Overview

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.

### Skip Counting

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.

### Factor Tree

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.

### Missing Factor

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.

### Division Problems

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.

Here is a link to 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

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.

### Initiation

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.

### Blank Templates

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.

### Accessing the Handout

Here is a link to the handout.

## Solving division equation proportions is challenging for two reasons. First, it involves fractions. Second, for some reason, many students struggle with solving division equations and a proportion is a more complex version. This post outlines a scaffolded handout to guide students through solving by multiplying both sides.

### Overview

The handout provides support in two ways. First, it draws upon prior knowledge of solving division equations Second, it scaffolds the initial multiplication. This post is in contrast to another in which I share scaffolding for cross multiplication, which is helpful if a variable term is a binomial.

### Review

Page 1 focuses on solving division equations to draw upon prior knowledge and to introduce the scaffolding. You can have students write a 1 under the factor.

### Proportions

This page uses the same scaffolding, but now with proportions. The focus is on multiplying the numerator, as was done on the first page.

### The Handout

Here is a link to the handout.

## 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.

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.

### Blank Templates

The second sheet has blank templates. One possibility is that you could assign students problems and they can complete the first 4 on here.

### Accessing the Handout

Here is a link to the handout.

## 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.

### Overview

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.

### Scaffolded Steps

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.

### The Handout

Here is a link to the handout. The image below shows page 3, a blank template.