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).

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).

A question was posed recently that I find intriguing and important. The question was, “what is the difference between slope, constant of proportionality, unit rate, and rate of change?” I researched the answer to this to evaluate my own understanding of these terms. Here is what I found (and am open to feedback as my understanding evolves).

First, I found a credible website that provides a glossary of math terms, the Connected Math Program page on the Michigan State University website.

In this glossary I found the definitions of the terms in question, along with the term rate.

I then found examples from a Google search that provided more of a visual image of each term.

Note that when we identify slope in an equation, we are identifying the slope of a line from the equation of the line. Slope is a measure of steepness of a line so the number likely should be thought of in that context.

The constant of proportionality can be found in different representations, but it is a constant while the slope is a measure of steepness – a special type of constant. (See definitions from MSU).

A rate, as seen in the definition above, involves units. A rate of change is the change of input and output variables so this is a little different than a simple rate as units may not be cited but could be cited (see last two images below).

The confusion for many of us is that there is overlap between these terms, depending on context. This is similar to the man on the right who can be referred as father or son, depending on the context. Slope is a rate of change for a line. A constant of proportionality is the slope for a linear representation of a proportional relationship.

The constant of proportionality is a constant but can be interpreted in a given context.

Slope is a ratio that can be interpreted in a given context. In the example below, slope = 100/1 and is interpreted as a rate of change of $100 per month.

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.

If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

I previously related elementary school word problems with math topics in secondary schools. The photo below shows a method to help elementary school students unpack the multiplication problem, to help middle school students identify the unit rate, and to help algebra students identify slope (you can focus on simple problems like this as an entry point to the linear function type problems).

In advance of this method, a review is conducted on the representation of multiplication using the groups of items model (below). By drawing a picture for the two parts of the problem that have a number, the students are guided to break the problem into parts and then to unpack the parts. The “5 boxes of candies” is represented by squares (or circles if you prefer) with no items inside. The “each box holds 6 candies” is represented by a single square with 6 items (dots) inside.

In turn, the drawing of the group of items leads to the multiplication statement, “6 candies x number of boxes.” I prompt students to include the items with the number as sometimes they will write this statement as “6 x number of candies”. I point out that 6 and candies go together. As seen in the previous blog post, the next step in this problem would be to replace “number of boxes” with the quantity given and then compute.

Here are excerpts from two handouts I use to help students understand how to write multiplication and rate word problems as math expressions. The image, below at top, shows a problem from the first handout I present. The students draw a single group represented by the rate expression (for elementary school word problems the term rate is not used). The image at the bottom is the same problem with scaffolding to write the multiplication problem. I find that students working on rates and slope in middle school, high school, and even in college struggle with this topic. I use this approach as part of a review of prerequisite skills before getting into rate and slope.

Top left is a scaffolding I use to help students learn to solve math problems using multiplication (3rd grade). The situations are typically rate problems (e.g., 5 pumpkins per plant or $2 per slice of pizza) although the term “rate” is not used yet. The same concept of rate plays out in high school with slope of a line, applied to real life situations (top right).

These types of problems start in 3rd grade (below, top left), play out in 6th grade (below, top right), into 8th grade (below, middle), and into high school algebra and statistics (below, bottom). I referenced this connection previously regarding word problems and dominoes. This highlights how crucial it is that strategically selected gaps in a student’s math education be addressed in context of long range planning.

Unit rate is an important topic in middle and high school. Unit cost (e.g., hamburger meat on sale for $2.39 per pound or you make $13 per hour) is an example of unit rate. This post shows how to use unit cost as an effective entry point to learning unit rates.

Significance

First, unit rates and unit costs are common in life. Second, in the Common Core State Standards math categories you can see that Proportional Relationships, and Ratios and Proportions (which includes unit rate) are a 6th and 7th grade topic and are then replaced by Functions in 8th grade. The proportional relationships are an entry point for functions.

Below is a photo showing a graph of a function. The slope of a line is the ratio of vertical change to horizontal change. In context, it can model the unit rate in a proportional relationship.

Unit cost can be challenging for students who need life skills math

Activity

First, I present a pack of items the student likes (4 pack of Muscle Milk for this student). Use a Jamboard to show a 4 pack and the price of the 4 pack (photo on left). Then I “pull out” the 4 individual bottles and divide the $8 among the bottles to show $2 for each bottle. Here are links to a FB Reel and a YouTube video showing how this works.

I Follow the same steps for ounces or pounds but show how 4 oz is divided into single ounces (in lieu of a pack divided into single items). Then the student shops for items that can easily be divided to get a unit cost.

Google Document

A follow up to the Jamboard is to have students use a scaffolded chart to shop for items. This will help them internalize conceptually what a rate and unit rate are.

Have them start with an item that is a pack. In the Google Document, they paste a screenshot, enter the cost and quantity, and then compute and enter the unit cost but use “for 1” before delving into volume and weight.

Accessing Jamboard

You have to make a copy of the Jamboard in order to use it.

61 cents per ounce is a rate of change. Graph the line modeled by this (y intercept is 0) and it becomes slope of the line. In referring to algebra we often hear, “when will I ever need this?” My response is “all the time!” Our job as teachers is to make this connection for students.

Slope is the rate of change associated with a line. This is a challenging topic especially when presented in the context of a real life application like the one shown in the photo. The graphed function has different sections each with a respective slope.

One aspect of slope problems that is challenging is the different contexts of the numbers:

The yellow numbers represent time

The orange numbers represent altitude

The pink numbers represent the slopes of the lines (the one on the far right is missing a negative)

Before having students find or compute slope I present the problem as shown in the photo above and discuss the meaning of the different numbers. What I find is that students get the different numbers confused and teachers often overlook this challenge. This approach is part of a task analysis approach in which the math topic is broken into smaller, manageable parts for the student to consume. Once the different types of numbers are established for the students we can focus on actually computing and interpreting the slope.

This instructional strategy is useful for all grade levels and all math topics.