Tag Archives: functions

Introduction to Unit Cost (Unit Rate)

Unit rate (e.g., hamburger meat on sale for $2.39 per pound or you make $13 per hour) is an incredibly important topic in middle and high school. First, unit rates and unit costs are common in life. Second, in the Common Core State Standards math categories you can see that Ratios and Proportions (which includes unit rate) are a 6th and 7th grade topic and are then replaced by Functions in 8th grade. Below is a photo showing a graph of a function you can see that the slope in an application is a unit rate.

The unit rate is also conceptually challenging whether it is in a function or is a unit cost at the store. This is a major sticking point for many students in special ed who have fallen behind. To address this, I used the approach below.

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. Finally, I have the student shop for packs of items at a grocery store or Amazon and compute the price for 1 item using a mildly scaffolded handout.

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.

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K-12 Math Education is a Line of Dominoes

Critical dominoes in math education start falling in 6th and 7th grade with the last ones falling in college. If you have a student who struggles with math and is entering or returning to middle school, now is the time to intervene to avoid more serious issues related to math education in the future. If your student is not going to college or is not accessing the general curriculum, I suggest you read this.)

Below is a chart showing the different categories of Common Core of State Standards (CCSS) math (called domains) at different grade levels. For the majority of students who will attend college, the traditional algebra based sequence (algebra 1, algebra 2, and maybe pre-calculus, calculus) is the path of math courses to be taken. Given this, for students who struggle in math but have a post-secondary education as a goal, the domains I emphasize in middle school are Expressions and Equations, Ratios and Proportional Relationships, and Functions. For high school, I emphasize Algebra and Functions.

Looking at the overviews for CCSS math standards (below) you can see the dominoes line up.

  • In 6th grade, Ratios and Proportions are an entry point for Functions in 8th grade which leads to Functions in high school.
  • In 6th grade, Expressions and Equations are the entry point for Expressions and Equations in 7th and 8th grade, which lead to Algebra in high school.

If your student is struggling with the middle school topics I cited and the gaps are not filled, the struggle will be carried with them into high school and into college.

I recommend the following:

  • Focus IEP math objectives on the priority units of the math curriculum, as cited above.
  • Ask for examples of mastery for the objectives to help you evaluate progress and mastery. Have this in place from day 1.
  • Focus on study skills, not just content mastery.

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Function Notation for Algebra

Below is a video of a lesson I recorded on function notation using the Explain Everything app. The lesson starts by addressing the concept of function notation by connecting it to the use of the notation “Dr.” as in Dr. Nick of Simpson’s fame. The lesson builds on prior knowledge throughout with a focus on color coding and multiple representations.

This videos shows an instructional approach to teaching function notation and concepts in general and video lessons can be used for students who miss class or who need differentiation.

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Function Notation

Screenshot 2017-11-24 at 8.24.37 AM

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.

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Functions Introduction Video Lesson

This video provides instruction to introduce the definition of and conceptual understanding behind algebraic functions.

Functions video


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Relations Introduction Video Lesson

This video provides instruction to introduce the definition and conceptual meaning behind algebraic relations.

Relations Video


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Graphs Representing Variable Relationships

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. van scatter plotThe 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. graph matching

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Kim and Kanye are Prior Knowledge (with scaffolding)

functions with Kanye

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.

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