Solving 2-Step Equation Scaffolded

Solving 2-step Equations is an escalation in terms of task demand. Starting in elementary school, they are exposed to math sentences in which there is a single operation. Now they are asked to choose a number to eliminate. This post provides details about a scaffolded handout. It addresses the choice of number to eliminate and how one choice may provide the path of least resistance.

The Steps

Similar to the handout for solving 1-step equations, this scaffolded handout engages the students with the mental steps in addition to the steps they write out when they show their work. A key concept is identifying that there are two numbers to eliminate and then eliminating them, one at a time.

Choosing the Number

In lieu of teaching the method of doing PEMDAS or order of operations in reverse, I focus on presenting both the respective steps for eliminating either number in the expression with the variable. I believe it strengthens their understanding of the steps for solving and it shows them a reason to choose one route over the other – but they are free to choose! Here is a FB Reel and Youtube video addressing this.

Access to Handout

Here is the handout.

Introduction to Scatter Plots with Google Sheets

This post outlines an activity to introduce linear functions (or scatter plots). The students are tasked with shopping for a used car – a specific make and model. They go to Carmax.com to find mileage and price for 10 cars for sale. They have to find a make and model that has at least 10 cars and can change the search radius to include all locations of Carmax as necessary.

They enter the data for each car into a table on a Google Doc. They are not to include the “k” or the “$” or “,” for price. This allows easier transfer of data. I do not use 0s for the mileage as the slope is more meaningful per thousand miles. For example, -$104 per thousand miles vs $.104 per mile.

Before they graph, you can provide them a common set of data to guide them through a trial run. This way you can show them your graph of the data to allow them to verify that they did it correctly. The data sets shown below are linked at the bottom of this post. (This can be useful for introducing systems of equations as Mustangs typically have a higher intercept and a steeper slope, which allows for a cluster of dots from both in an intersection.)

They copy and paste the mileage and price into a Google Sheet and attempt to graph. You can provide a link to a YouTube video on graphing a scatter plot to free you up to help individuals. The title of graphs should have the variable(s) and the individuals under study. A subtitle can be included to show when data was collected or a data set was accessed. The variables should include units.

A next step would piggy back off of this activity with a Jamboard addressing mileage and price to help students interpret scatter plots.

Here are links to items used for this activity.

Unpacking Graphs Using a Rollercoaster

I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).

Handout

Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.

I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.

The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.

Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.

The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.

I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.

Jamboard

Here is a Jamboard. You have a make a copy to access it (see photo at bottom). Here is a FB Reel and a YouTube video showing how it works.

Intro Linear & Proportional Relationship

This post provides a handout to introduce linear functions and proportional relationships. Multiple instructional strategies are implemented. There is context that draws upon prior knowledge, scaffolding, and visual representations. Graph and table representations of the functions are used as an entry point, without the use of equations.

Overview

There is a difference between proportional relationships and linear functions, which is addressed in another post. Proportional relationships are a subset of linear functions. They can be used as an entry point by citing it as prior knowledge and then showing how they have a constant rate of change.

Proportional Relationship

The students complete the table with the images then the table with the variables. This leverages the context to help them make sense of the table and graph. You can follow up by asking them the total for 0. This allows you to highlight the intercept.

Linear Functions

Linear functions are introduced in similar fashion, including with 0 toppings for the intercept.

Handout

Here is a link to the handout.

Introduction to Intercepts – Mini-lesson with Scaffolded Section for Computing

Here is a link to the document, with images showing the notes. This is a mini-lesson with the following components.

  • A fill in the blank for writing the lesson objective.
  • A Do now which serves as an initiation to the lesson. The y-intercept can be discussed in the context of buying 0 slices of pizza and paying $1.
  • A notes section on what an intercept is.
  • Practice session on identifying intercepts in graphs and tables.
  • A scaffolded steps section on computing the ordered pair of the intercepts.

Graphing a Quadratic Function in Vertex Form – Scaffolded Handout

Here is a scaffolded handout. The first problem has additional scaffolding to convert from addition to subtraction and vise versa. The first problem does not require the transformation of the operations but the second one does.

Completing the Square – Scaffolded Handout

Here is a link to a scaffolded handout with all the steps, including the step to convert the lead coefficient to 1. The scaffolding is complex because the math topic is complex. There is an example showing how to use the document.

Intro to Solving Equations – Seesaws and Oranges

Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions.

Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.

Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.

The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a post with an updated version of this handout.

Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).

Make a copy of the Jamboard so you can edit it.

Proportional Relationship (with k) vs Linear Function (with m)

There have been interesting discussions on various Facebook Teacher pages about proportional relationships vs linear functions. This mirrors discussion about the constant of proportionality vs slope vs unit rates.

The difference between the proportional relationships and slope is context and the ratios. The ratio for the former addresses the variables themselves. The ratio for the slope addresses the change in the variables. This arises from context. To flesh this out let’s use the pay as a function of hours, with $15 an hour for the hourly rate.

  • If we focus on the fact that to compute the total pay, we multiply by the hours worked by 15 we have a proportional relationship and the $15 an hour is a constant of proportionality.
  • If we focus on the fact that every increase of 1 hour results in an increase of $15 in our total pay, we have a linear function and the $15 an hour is a rate of change (or slope of the line). Because of the context, we have different constants (k vs m).