This video explores why variables are worth exploring in-depth with students. We frame some common misunderstandings about variables and ways to address them.
Write, read, and evaluate expressions in which letters stand for numbers.
CCSS.Math.Content.HSA.SSE.A.1.a Interpret parts of an expression, such as terms, factors, and coefficients.
Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
de Araujo, Z., Dougherty, B., & Zenigami, F. (2018). Putting essential understandings of equations, expressions, and function into practice in grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
In our video titled “What are variables?”, we left you with this definition: Variables are symbols that represent unknown or varying mathematical objects. Now we’re going to discuss some student stumbles that are likely to occur as students work with variables.
But first, doughnuts — an example about doughnuts, that is.
Harvey’s Doughnut Shop sells 6 doughnuts in one box. How do you think students would represent this situation algebraically? They will often write 6d = b. This is not correct, though, because this equation implies that only 5 doughnuts would be in 30 boxes (show the animation).
So if this isn’t correct, why do students come up with it at all?
The short answer is, students are thinking of d and b as labels, not variables. They’re imagining this: (show 6 doughnuts equals box visually then underneath put in the symbols). This makes sense because when we say things like 100 centimeters are equal to 1 meter we write it like this (100cm=1m). Here, cm and m are labels, not variables. Students are bringing this same idea to the doughnut shop, where it doesn’t belong.
In the doughnut example, d should represent the number of doughnuts and b should represent the number of boxes, so students should produce an equation that is equivalent to this (show d/6=b). So how do we help students get here?
Have students talk through the scenario in as many different ways as possible. They might say things like,
- 6 doughnuts come in a box
- 1 box has 6 doughnuts, so 2 boxes has 12 doughnuts
- 3 doughnuts would only be a half a box
- For every 6 doughnuts, I fill one box
- 10 boxes is 60 doughnuts
Let them play around with it and maybe even create a table to keep track. Then, as they create their algebraic equations, they can go back to the various phrasings and check if their equations are consistent with them.
Don’t use d and b for doughnuts and boxes! Learning experiments have shown that using the initials tends to perpetuate students’ misuse of variables as labels. This is because it matches our speech rather than the mathematical structure. Think about it — 6 doughnuts equals a box. If I were shortening this linguistically I could just say 6d=b, but that doesn’t have the right mathematical meaning. If instead we ask students to use random letters like m for the number of doughnuts and q for the number of boxes, then, students are less likely to simply draw on the words and instead they will concentrate on the quantities, as intended. 6m=q doesn’t have direct linguistic translation to doughnuts and boxes, so they’ll have to think about the mathematical relationship, which is that there are actually six times as many doughnuts as there are boxes.
So be brave, try it out, and let us know how it goes in the comments.