## This post provides details about an artifact that has a manipulative and visual representation of tax rate and discount rate. These contexts are used as an introduction to percent change. The manipulatives are presented on a Google Jamboard.

### Overview

The price or original price is presented as dollar bill. The bill is cut into proportional pieces to show the increase or decrease amount, visually, as a part of the original amount. The pieces can be moved around the Jamboard and replaced by other denominations.

### Slides

The slides are presented in the slide show below. They are arranged in the following order. the slides show the different positions of the manipulatives, e.g., how the \$20 bill is cut into discount and sales price.

• Slide 1: 5% Tax Rate for \$20 price – compute the total to pay
• Slide 2: 20% off discount for \$20 original price – compute the sales price
• Slide 3: generic tax rate
• Slide 4: generic discount rate

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

## 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.

## Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.

### Visualizing Unit Rate

Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.

This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.

The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.

### The Handout

Here is a link to the handout. It is in WORD format to allow you to enter your own problems.

## Intro to Ratios using Jamboard

Below are images from a Google Jamboard for a hands on introduction to ratios. (See image at the bottom for how to make a copy in order to use it.) The images are from Clever Cat Creations and provide a visual representation. The moveable items engage the students kinesthetically. It also helps unpack the concept of ratio as a comparison of two quantities as the students count out the quantities and represent them as numbers in a ratio. The scaffolding guides the process.

First, students move the terms to make a connection between the statement and the ratio.

Then the objects are counted and moved.

Then the ratio is written.

The quantities can be flipped to show an alternative ratio.

There is a blank to create your own and another with shapes.

You have to make a copy in order to move the pieces.

## Solving Proportions – Scaffolded and Jamboard

Below are images from a Jamboard and a handout that scaffold cross multiplying to solve a proportion. (See image at bottom to make a copy of the Jamboard.) This is an entry point, with a focus on how to write the ensuing equation. Solving would be a prerequisite skill so it is not addressed (but obviously would follow). This allows for less task demand placed on the students and for more time spent on the new steps.

The arrows and shading scaffold the cross multiplication step. Students move the terms from the proportion to the equation. This allows for kinesthetic engagement and helps students see how the equations are formed. The scaffolding for the equation guides students to writing the equation, which I have found a challenging step for some students. The equation is written first as factors to reinforce the idea of multiplication, then the students simplify for the second equation.

The handout draws upon the Jamboard and uses the same scaffolding. The template is blank to allow for use with other handouts. The students can copy problems from another handout and follow the scaffold to get to the equation. The steps and equation can be transferred over to the handout.

Make a copy of the Jamboard to use 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).

## Intro to Ratio Tables

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

To introduce ratio tables, I draw upon a relevant prior knowledge for a child. Food, especially pizza is a go to context for me.

Use a CRA approach by using manipulatives (concrete), pictures (representational), and numbers (abstract). The ratio can be changed and other contexts can be used (e.g., \$3 per slice and use dollar bills and the slices).

Here is a link to the Jamboard for an intro to ratio tables. You can make a copy.

## Group Model for Proportional Relationships and Slope Application

If you find this helpful, maybe consider making a small donation to help build an accessible playscape for children with disabilities.

In my experience, many students struggle with translating word problems modeled by a proportional relationship or a linear function into a algebraic expression or equation. (Image below.)

Here is a link to a Jamboard that can be used to introduce the algebraic expression to model the unit rate. (See image below on how to copy and edit.)

Here is how you can use this to introduce modeling the word problem.

• Start with the unit rate concept. In this case there is \$45 “in” every hour. This is modeled in slide 1 (top 2 photos).
• The next 2 photos show slide 2 in which the student duplicates the \$45 image and fills 2 hours, with \$45 “in” each. They complete the multiplication expression by multiplying by 2.
• This is followed by the same steps for 3 hours (photo bottom left) and sequentially to 6 hours.
• In the last slide there are no hours shown because the # of hours is unknown. This leads to using “X” to represent the unknown NUMBER of hours (I don’t let students get away with x=hours) and finally the algebraic expression (bottom photo).

You can make a copy and edit it.

## Unit Cost and Actual Shopping

I previously shared that grocery shopping has a lot of tasks that are overlooked. One is working with unit costs. There are two math tasks related to unit cost, interpreting what a unit cost is and computing the total cost for buying multiple items.

When I take a student into a grocery store to work on unit costs, where is what I do. I start with a pack of items (photo below) and ask the student to compute the cost for 1 item, in this case, “what is the price for the pack of chew sticks, how many in a pack, how much for 1 chewstick?” Then I prompt the student to compute the total if he buys 3 packs. This allows the student to differentiate between the two tasks.

The cost per items is easier to grasp and then is followed with the same prompts for a jar of sauce (below). I have the student compare unit costs for the large vs the small jars and ask, “do you want to pay \$4.99 per ounce or \$5.99 per ounce.” This language is more accessible than “which is a better deal?” You can work towards that language eventually.

These tasks can be previewed at school with a mock grocery store. The price labels can be created on the computer.