Functions are perhaps the most prevalent and important topic covered in secondary math, aside from maybe 1 variable linear equations. The concept of a mathematical function is challenging for many students. This post provides details about a meaning making approach to introducing functions.
The introduction is presented on a Google Jamboard, to allow for movement in the pairing of inputs and outputs. It starts with analogies pairing of items using a gumball machine and a Coke machine and proceeds incrementally towards the various representations. The functions are contrasted with examples of relationships that are not functions.
Slides of the Jamboard
Slides1 and 2 present the gumball and Coke machines. Students can move the items to see how a quarter can result in 2 different color gumballs while the Coke button results in only 1 output.
In slides 3 and 4, the use of an hourly wage introduces input and output with quantities. Slide 4 shows two different pay amounts for the same number of hours worked. This taps into prior knowledge.
The sequencing progresses through
Each includes an example and a non-example.
The last slide provides a sorting activity.
Access to Jamboard
Here is the link. To access the Jamboard, you need to make a copy.
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.
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
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.
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.
You have to make a copy of the Jamboard in order to use it.
Function notation is challenging for many students yet we teachers overlook the reasons for the challenges. For example, students see the parentheses and rely on the rule they were taught previously, y(5) is “y times 5”. Because this is overlooked by teachers we often skim over the concept of notation and delve into the steps. This document is a means of presenting the concept and the use of function notation in a meaningful way. Feel free to use or revise and use the document as you wish.
Students often struggle with the jump from 1 variable situations to 2 variable situations, i.e. functions. Visuals are an effective way to present concepts which may be abstract, e.g. functions. Graphs and visuals in general are designed to allow a big picture view of a concept. The photo above shows mileage and price for Honda Odyssess vans collected in January 2012. The work is from a student who has a non-verbal learning disablity (basically a mild version of asperger’s). Students are first asked to identify the van with 66,000 miles and that costs $21,000. This ensures they understand what is presented. Then they are asked a leading question about mileage and price (#3). Finally, in question 4 they are asked to use the graph to draw a conclusion – choose the best van (based on mileage and price). This leads into a discussion about the relationship between mileage and price (you can see notes I wrote on the graph to facilitate this).
The photo below shows four scenarios and one of four graphs students are to use for matching. For the first graph students were to determine that the graph best models the US economy (problem given in 2013). Over the past few years the economy dropped rapidly but has been improving since. This student (same as the one identified above) identified one aspect of the graph – the recent rise. Breaking a graph down into parts and comparing the parts is an example of Bloom’s analyze level. By justifying his choice of C this student was evaluating.
This is a portion of scaffolded notes I provided for a lesson on functions. This shows two key strategies I often employ: scaffolding and connections to prior knowledge.
The scaffolding is seen in how blanks are provided for students to fill in key information. This saves time on copying notes while still engages students in note taking. In the notes handout I include photos that enrich the notes.
The prior knowledge is the photos. The guys are Kanye West and Chris Humphries (basketball player) with Kim Kardashian (dark hair) and Amber Rose. Kanye was dating Amber and Chris was married to Kim. Kanye then cheated on Amber with Kim. Most students fully understood this situation which allowed for carry over into the concept of functions. While this connection is not concrete as in CRA representations, it does make the concept more concrete for the students.
Note: a visitor asked if this presentation sent a message to females. That’s a fair question. My response is that the natural follow up is to show Kim matched with Kanye and Chris and ask if that is a function.