The Language Trap: Decoding "More Than" and "Less Than"

If you’ve ever seen a student read the phrase "5 more than x is 12" and immediately write $5+x=12$, only to see them do the exact same thing for "5 more than x is greater than 12," you aren’t alone.

For many students, word problems are less about logic and more about "keyword hunting." They see "more than" and instinctively reach for the plus sign. They see "less than" and prepare to subtract. The challenge isn't that they don't know the math; it's that they don't recognize the grammar of inequalities.

Here is how to help students distinguish between an operation (addition/subtraction) and a relationship (inequality).

The most powerful tool in a student’s arsenal is the word "is." In the English language, "is" acts as a bridge to a comparison.

• The Operation (Action): "Six more than a number."

  • There is no "is." This is an incomplete thought, a mathematical phrase. It translates to $n+6$.

• The Inequality (Relationship): "Six more than a number is greater than ten."

  • The "is" changes the "more than" from an instruction to add into a statement of comparison.

The Strategy: Have students circle the verb in every word problem. If they find "is," "was," or "will be" attached to the comparative phrase, they are likely dealing with an inequality or an equation, not just an expression.

When students think of "more than" as addition, they are thinking of a destination. When they think of it as an inequality, they need to think of a region.

Ask your students: "If I have more than $5, do I have exactly $6?" The answer, of course, is "Maybe, but I could also have $100."

By using number line sketches in their journals, students can visualize the difference. An operation is a single point moving forward or backward. An inequality is a shaded arrow that covers infinite possibilities.

Teach students to look for limiters. Words like "maximum," "minimum," "at least," and "budget" are red flags for inequalities.

• Addition context: "Sarah has 5 apples and got 3 more." (She is combining items to find a total).

• Inequality context: "Sarah needs more than 5 apples to bake a pie." (5 is the threshold, not a part of a sum).

Give students "Switch-Up" drills. Provide two nearly identical sentences and ask them to write the mathematical equivalent for each:

1. "A number decreased by 10." ($n-10$)

2. "A number is less than 10." ($n<10$)

By placing these side-by-side, students begin to see that the "less than" in the first sentence is an action being performed on the number, while the "is less than" in the second is a boundary the number cannot cross.

Moving students away from keyword hunting requires us to teach them to be "math linguists." When they stop looking for "more" and start looking for the relationship between the values, the confusion between $x+5$ and $x>5$ evaporates. It’s not just about the numbers; it’s about what the numbers are allowed to be.

The relation between compound inequalities and absolute value inequalities

 I finally ended up using a textbook that I like.  It is older but it groups many topics together so they flow well from one to another.  One such flow is the book teaches how to solve compound inequalities first and in the very next section, students are asked to use what they learned with compound inequalities to solve absolute value inequalities.  

I think this made it so much easier for my students because they could "see" why these inequalities had to be solved as two problems rather than one due to the definition of an absolute value. I was able to relate the AND's and OR's to the appropriate absolute value inequality even to the point of showing how can be solved the same exact way as you do for a compound AND statement.

It was nice to have the book actually teach absolute value inequalities by having students rewrite them into compound inequality equations after reducing any into the absolute value being greater than or less than to a value. When using the idea of rewriting the absolute value inequality into a compound inequality, most of my Algebra I students were nodding and it was like a light bulb going off in their heads.

The biggest issue they had was to remember that the absolute value inequality with a less than sign was solved using the AND while the greater than required the OR situation to solve.  In addition, I had to remind them about the special cases of no solution or all possible solutions because they got in a rhythm solving the inequalities.  It's the same as when students first solve inequalities and just whiz through one that equals a negative number.

Once I finish this section, I should take time to have students convert compound inequality problems into absolute value inequalities so students better understand it is a two way relationship.  The previous section did not take time to show how to do that so they only do the conversion one way.  I think I'll find a worksheet of mixed compound and absolute value inequality problems so they can practice going from one form to the other.  I don't know if I'll actually have them solve the problem but I want to have them comfortable going from one to the other.

The longer I teach, the more I realize how important it is for students to make connections between one thing to the next.  I've expanded how I teach things so I am introducing ideas before we actually get to them or add depth to a topic as I teach it.  This is one topic, I need to show to students.  Let me know what you think, I'd love to hear.  have a good day.

Real Life Inequalities.

  I decided it was time to have my students do  bit of the work so in Algebra I, I began one variable inequalities by having students find real life examples and write them on the board.

We run into them all the time but we don't connect the situations with being inequalities.  One such situation deals with posted speed limits.  You can go as fast as you want up to that speed but not over it or you might receive a ticket.

Another two situations involve credit cards.  The first is with the credit limit.  A person can charge up to that amount but no more so the inequality might read x < $10,000.  The other inequality is in regard to the minimum payment.  That is the least amount a person can pay but they are welcome to pay more.  The equation might look like x > $122.30. 

Twitter uses an inequality when they restrict messages to 140 characters while many contact forms might restrict the message to 500 characters.  Then there is the issue of fundraising.  Most groups set a minimum amount they hope to raise through selling candy, cookies, or a raffle.  Although they hope for minimum amount, they will not stop there.  Instead, they will continue accepting donations.

There are some jobs out there which require you to sell a minimum amount in order to receive additional funds.  It might be you must sell over $8000 worth of computers before you get a bonus.  This would be another real life example of an inequality. 

In addition, there are other examples such as elevators which have a maximum weight load usually couched as a maximum load of 9 people or 1000 pounds.  Check the bridge signs and they might say have a restriction on maximum truck weight or a height restriction so a vehicle must be under 7 feet off the ground.

There is also the minimum travel time listed when driving or flying.  Out here the minimum is 45 minutes from Bethel to the village if everything is great and you get the fast plane but it could take 2.5 hours if you end up going to the two other villages first.

Sometimes, when we find something we want to buy that special pair of shoes, or dress and we have to plan the amount of money we need to save each week.  Or when you buy a house or a car, you have a minimum payment to make each month.  My mother always said that if you pay $100 extra each month, you'll pay your mortgage off much sooner.  I've always kept that in mind.

So many different possibilities, all provided by my students the other day.  It allowed them to find the real life applications before we began studying the topic so they can build a solid foundation.  Later in the week, I'm going to give them an inequality and they will have to write the situation to go with it.  I'll let you know how that goes.

Let me know what you think.  I'd love to hear.

Real Life Applications of Systems of Inequalities and Inequalities.

In a few weeks, I'll be teaching solving systems of inequalities and one of the first questions a student will ask is "How is this used in real life?"  I always struggle to answer this one because I got my degree in theoretical math so I'm great with the math but not so good with the practical. 

This is used to determine a solution to situations such as figuring out the number of a product that should be produced to create the most profit or determining the correct mix of drugs for a patient.  Imagine its used in medicine.

Its also referred to as the Theorem of Feasible Regions.  In other words, it is the set of coordinate pairs that solve a systems of inequalities.  Its the region which satisfies restrictions placed in linear programming.

So then what does an inequality represent in real life.  In real life, an inequality is recognized by the use of limits such as a speed limit of 75 mph, a minimum payment on your credit card, limit of text messages, or time needed to travel.  For any of these examples they are actually inequalities. 

The speed limit example means you are not to travel above 75 mph but you could easily travel below it.  When you receive the credit card bill, they say you can pay $50 but you can pay more.  This is only a minimal suggestion so your payment is $50 or more.  Many people choose a plan that says they cannot send more than 250 messages in a month or often we calculate our trip to or from someplace based on time. If everything is right, it takes me a minimum of 5 minutes to walk to work but if the weather is bad, it could take 10 to 15 minutes.

In addition truckers face inequalities all the time when they cross bridges and have to keep track of their weight.  An example might be a bridge can only handle trucks whose weight is not over 65,000 pounds.  A trucker has to know the weight of his truck and trailer so he knows if he can use the bridge or must plan an alternate route.

Back to driving but not the speed.  Another inequality has to do with obtaining your drivers license because in most states you must be 16 in order to get one.   This would be a x is greater than or equal to situation but you might have to be 18 to get an unrestricted license.  Along these same lines, there are minimal ages for buying liquor or cigarettes.  Both of these are inequalities.

There are also thermostats in the cars that operate on inequalities also with voltage regulators and even Body Mass Index.

Its easy to find single examples of inequalities but not for systems of inequalities unless you look into linear algebra or linear programming.  At least I have a better idea of how to explain the use of systems of inequalities.  I hope you learned some things because I know I did. 

inequalities and graphing apps.

In algebra 2, we are learning to graph linear and absolute value inequalities using a graphing app.  I like the free graph calc for graphing as it has so many choices and resembles a regular graphing calculator.  It does not seem to support graphing inequalities which is fine with me because the students can graph the base linear or absolute value equation, then do the analysis to determine if they need a dotted or solid line and which part is shaded in.  Too often my students just copy down the result from the graphing calculators or a regular calculator without checking to see if its reasonable. 
Yesterday, all the high school teachers gave students a directions test during study hall. The test had them writing down things like the letter D in the middle of the page, writing down their favorite musical group in a corner, etc.  We did this because the majority of our students do not read directions.  As expected, most students failed the test but the kids got such a charge out of the test.  What they don't know is that they will get another one next month.
Finally, the tech dept and I are still working to figure out how I can have students either send the material to me using e-mail or get it to me via a drop box type of situation.  I am not allowed to use google docs, regular e-mail accounts, drop box or any other type of app like that.  So it is a challenge.  I am doing some research and have a couple possibilities that I need to explore hopefully over the weekend.  I hope by January to greatly reduce the paper flow.