Systems Word Problems Scaffolded

Word problems are challenging for many students. Writing a system of equations to model a word problem has unique challenges. This post provides details about a scaffolded handout with color coding can unpack the process for generating the appropriate system of equations.

Unpacking the Word Problem

A mistake I have witnessed over the years is students mistakenly using given values for both equations. In the problem below, students are far more likely to generate the equation for the yellow part: 2x + 3y = 24. The challenge is that the blue part has only 1 number so students will often write 2x + 3y = 10, using the dollar quantities a second time.

By highlighting the two parts of the word problem with given values, the students can match parts of the word problem with respective equations. The scaffolding separates the parts, and the color allows for matching.

Solving

The rest of the problem is prior knowledge with the students using one of the methods for solving. The scaffolding continues to lower the task demand by reducing the need to remember all the steps. This allows them to focus more bandwidth on the new steps.

Access to Handout

Here is a link to the handout.

Power Rules on a Google Jamboard

Exponents and Basic exponent rules are challenging. The Power Rules add another layer of challenge. This post outlines an instructional approach. The original problem is decomposed and then recomposed to show how the underlying concepts of the Power and Power of a Product Rules.

Overview

The Jamboard is configured in similar fashion as the Jamboard used for the Product and Quotient Rules. The exponential terms and variables are moveable parts. The background is a scaffolded to guide the decomposition. Here is a FB Reel and a YouTube video showing how it works. NOTE: I decompose the expression down to individual X values in lieu of using the Product Rule because I want them to see how many Xs there are. Also, the Product would be relatively new to them, I wanted to reduce the task demand placed on the working memory.

Jamboard Access

Here is a link to the Google Jamboard. To get access, you must make a copy.

Variable Both Sides Equations Scaffolded

Solving the variable on both sides is the Cerberus of 1 variable linear equations. It has multiple steps, simplifying expressions, and eliminating a variable expression. The later is a new step, added on to all the other steps. This post describes a scaffolded handout to guide students through the mental steps and the written steps.

Overview

The process starts with mental steps of identifying the two variable terms. This directs students to focus on identifying that the equation has a variable on both sides which in turn leads them to understand the algorithm they will follow. The circling focuses attention on the operations. Then the students choose which variable term to eliminate and identify the inverse operation. The written steps are then guided.

Choosing the Variable Term

As I did in the scaffolded handout for solving a 2-step equation, I have students solve the equation two different ways. This time by eliminating each variable term respectively. This allows them to see for themselves which term may provide the path of lesser resistance.

Accessing the Handout

The handout can be found here.

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.

Exponent Rules with Jamboard

The Product and Quotient Rules are challenging largely because there are few ways to present them in an accessible format. This post outlines an approach using manipulatives on a Google Jamboard.

Jamboard

The Jamboard activity involves using moveable variables as manipulatives to provide a hands on approach to learning the rules. The first slide presents the idea of exponents as repeated multiplication. The Product and Quotient Rules use the manipulatives approach to unpack the underlying concepts of the rules. Note: there are set problems followed by blank templates. Here is a FB Reel and a YouTube video showing how this works.

Accessing Jamboard

Make a copy to access the Jamboard

Solve Equations with Like Terms Scaffolded

Many students struggle with solving equations. Many struggle with simplifying expressions. Putting these together becomes an algebraic version or Orthrus (brother to the more famous 3-headed dog, Cerberus). This post shares a scaffolded handout to guide students through the process while making sense of each step.

Explanation of Handout

The first two are mental steps. By addressing them explicitly with students providing a written evidence of their thinking, the mental steps can be observed and assessed. The operations are circle to be proactive in addressing common misconceptions. This can invoke a discussion on the meaning of the operations negative and subtraction. Once the expression is simplified, prior knowledge of 2-step equations kicks in. Here is a link to a previous handout.

The handout link.

Introduction to Adding Integers

Operations on integers and integers in general is challenging for many students. Negative numbers are abstract. Whole numbers and fractions can be represented with images. The activity presented draws upon student prior knowledge of thumbs up and down in a vote to make negative more accessible.

The following images are from a Jamboard. Here are accompanying videos on FB Reels and Youtube showing how this works. There are 3 sets of images (or chunks) loosely following a CRA appoach and all referring to the same two situations.

  • Prior Knowlege drawing upon a classroom setting (concrete)
  • Transition using thumbs (representational)
  • Introduction of adding integers, using thumbs (more abstract but still supported by

Prior Knowledge

The activity starts with a couple of classroom votes using thumbs up and down.

This is followed by scaffolding to focus on how the voting works through a comparison of the quantity of thumbs up vs down.

Transition

This section introduces thumbs as counters for integers, which is a common instructional strategy (yellow for positive and red for negative). The scaffolding is the same.

thumbs from Educlips on TPT

Adding Integers

Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.

Accessing Jamboard

To access the Jamboard you must make a copy.

Customized Number Lines for Handouts

Per request, I created a short video showing how I create customized number lines on WORD. This post also includes a link to a WORD document with 3 customized number lines: time, money with negatives, and miles.

Elapsed time

The image below is from a post on elapsed time. I wanted to create different time scales to match clocks I could create on math-aids.com.

The Video

In the video I show how I created the time number line. In the top image below, you can see the table highlighted. I then show how I copy and paste the number line and then edit to create units with money, with negatives.

Here is a screenshot of the video. You can see the number line in an early stage of development. Below the image is a link the video.

Link to video

Handout

Below is an image of the three customized number lines. Here is a link to the handout, which is in WORD format to allow you to revise to suit your work with students.

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Elapsed Time with Number Line and Clocks

The images below are from a handout to introduce elapsed time. This a revised version of another handout I created. The sequence in chunked to incrementally present additional elements. A number line is used to model, first on Jamboard then on a handout, then clocks are introduced. The first problem has an exact hour on the second clock to make it more simple but to still include minutes.

The clocks were created on math-aids.com, which has a page to allow you to choose times to be represented on clocks. They create clocks with color coded hands, which I follow with highlighters on the handouts and Jamboard.

First, the identify the the upcoming whole hour and marks the hands with highlighters or colored pens or pencils.

Determine the number of minutes to the hour.

Identify the whole hour preceding the second time and marks accordingly.

Determine the number of minutes from the whole hour to the second time.

Use the green marks used to identify the whole hours and determine how many hours passed.

I did not create a spot to write the answer to cut back on visuals.

The first page provides an introduction to the use of the number line without having to process the clocks.

Mark the whole hours.

Determine the number of minutes preceding and following the whole hours.

Determine the number of hours that passed.

A Jamboard is used to model the first 4 problems to engage the students kinesthetically and to unpack the concept. The students can do a Jamboard slide then work on the matching problem on the handout. (See photo at bottom for access.)

On the handout, I addressed the minutes of both clocks before determining hours. The Jamboard person can be used to flesh out the concept of time passing as the person walks. As a result, I suggest determining the hours before the minutes on the second clock as the person walks the entire way. When you return to the handout, you can reference the person walking the last 10 minutes and even show the students the Jamboard again when you do those minutes before determining hours.

Make a copy of the Jamboard in order to use it.

Intro to Factoring Out GCF – Scaffolded and Jamboard

The post shows a lesson to introduce factoring out a GCF. The handout is chunked to incrementally add task demands. A Jamboard is used as a complement to flesh out the concept of a factoring by allowing engaging the student kinesthetically, with scaffolding as guidance. (See image at bottom for how to make a copy to access it.)

The sequence starts with a review of factoring and distributive property. The boxes are a preview of the scaffolding for the full factoring to be presented in subsequent chunks.

The Jamboard slides aligned with the first chunk has students pull apart the variables with exponents to help them understand the meaning of the exponents during factoring. For example, there are two Xs overlaid on the x squared. Similarly, there are two 5s, one overlaying the other to allow them to be distributed.

The next chunk in the lesson is factoring out with the terms written with the multiplication symbol written in between. The problems are aligned with the distributive problems to help with the concept of “reverse distribution.” The same problems are presented again in box form as the boxes will be used to support the full factoring. Using the same problems helps the students understand how the boxes work. The Jamboard reinforces the distribution by allowing the movement of the 5s out of the binomial, including with the box method.

The last chunk combines the factoring of individual terms and the factoring out. This is conducted with the box methods for support, and then the boxes are faded. The scaffolding starts by writing the expression with parentheses because that step is easy to overlook when explaining it to students and can be confusing for students.

Here is what the final version looks like.

Note: the handout has blank templates for copying and pasting if you want to create your own handout.

Make a copy of the Jamboard in order to access it.