Link to a YouTube video showing how it works
Link to a Google Slides presentation showing the steps. Make a copy to use it.
Link to a Facebook Reel showing how it works.

Link to a YouTube video showing how it works
Link to a Google Slides presentation showing the steps. Make a copy to use it.
Link to a Facebook Reel showing how it works.
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.
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.
Here is a link to the Jamboard. You must make a copy to access it.
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.
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.
This page uses the same scaffolding, but now with proportions. The focus is on multiplying the numerator, as was done on the first page.
Here is a link to the handout.
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.
Here is a link to the handout. It is in WORD format to allow you to enter your own problems.
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.
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.
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 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.
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.
You can make a copy and edit it.
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.