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Which Is Bigger?

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem requires students to appreciate the significance of variables in an algebraic expression. Working on the challenges will offer students the opportunity to apply their understanding of equations of straight lines and simultaneous equations. The problem could also be used in preparation for work on inequalities.

Possible approach

Start the lesson by posing the question:
"Which is bigger, $n+10$, or $2n+3$?"
 
Give students a short amount of time to decide, and then ask them to discuss the justification for their answer in pairs. Look out for any pairs using a graphical argument to support their conclusions.
 
Share these discussions as a class. (If everyone agrees that one particular expression is bigger, use Charlie and Alison's example in the problem to generate some controversy.) 
 
"Is there any way we could represent what's going on visually, to convince ourselves that the first expression is bigger when $n< 7$ and the second expression is bigger when $n> 7$?"
 
Once there is an understanding that comparison of the expressions depends on the variable $n$, and that the comparison can be done graphically, set the next task:
 
"For each pair, can you work out when each expression is bigger?"
$2n+7$ and $4n+11$
$2(3n+4)$ and $3(2n+4)$
$2(3n+3)$ and $3(2n+2)$  
 
Again, give students time to work on this in pairs, making sure they are ready to justify their answers using the insights they have gained.
 
Finally, set them to work on the challenges offered in the problem. One nice way to round off the task could be to set up a graph plotting program (Geogebra is available to download for free) and ask each pair of students to read out the expressions they have found. As the expressions are plotted, the class can quickly decide whether they satisfy the requirements. This helps to capture the idea that there are infinitely many sets of expressions that satisfy each condition.

Key questions

Is one expression always bigger?
How can you decide when each expression is bigger?

Possible extension

Introduce challenges that require quadratic expressions as well as linear ones.
For example:
"Can you find two expressions so that the first is bigger for $n< 0$ and $n> 3$, but the second is bigger when $n$ is between $0$ and $3$?"

Possible support

Parallel Lines may be a suitable preliminary task for students who are not yet confident at working with equations of straight lines.