10 Math Problem Solving Activities for Middle School

Looking for some math problem-solving activities for middle school? Good, you’re at the right page then.

Right before children enter Middle School (around the age of 11 or 12), they enter a critical developmental stage known as Piaget’s fourth and final stage of cognitive development.  It’s at this stage that children demonstrate marked growth in a number of areas, ranging from making hypotheses and inferences to thinking abstractly and using advanced reasoning skills.  In line with this crucial phase of a child’s development, Middle School Math curricula are designed to stretch the bounds of adolescent thinking while also helping them to establish new skills and sound mathematical habits.

One way that educators try to ensure this is through common core standards that can be applied to Middle School-aged students.  These standards seek to achieve eight distinct objectives, which help foster the developmental transition addressed by Piaget.  The objectives:

1. Make sense of problems and persevere in solving them;
2. Reason abstractly and quantitatively;
3. Construct viable arguments and critique the reasoning of others;
4. Model with mathematics;
5. Use appropriate tools strategically;
6. Attend to precision;
7. Look for and make use of structure;
8. Look for and express regularity in repeated reasoning

allow for a lot of leeways as well as creativity in the way that problems are both presented to and solved by students.

The first objective, for example, emphasizes a student’s ability to not simply apply an algorithm to a problem, but more pointedly, make a decision and implement it.  This process can draw out drastically different reactions in different students.  For some, the prospect of being creative and innovative in thinking of ways to solve brain-bending problems is exciting, and often even addicting.  On the other hand, getting past the roadblocks that come along with solving a tough problem can be frustrating and, at times, discouraging for students.  It is in these moments that establishing math skills that promote perseverance are most critical.

A quick and easy way of avoiding that anticipated frustration that students might encounter in the face of challenging math problems is equipping them with an arsenal of tools and approaches through which they can tackle such problems.  

If, for example, you told me that I was a bird with a short, stubby beak that had to find a way to drink water from a glass that was only half-full with only a pile of stones at hand, I might get frustrated pretty quickly upon realizing that my beak did not reach far enough down to allow me to drink.  I might peck a few times in vain but would remain parched.  

With the right set of dynamic problem-solving skills at my disposal, however, I might think of the problem in a different light, and realize that by dropping enough stones into the glass, I could make the water level rise enough that my beak could easily extract all the water I desired.

Applying these kinds of problem-solving skills to questions that are appropriate for Middle School students can fortify grit, the quality of not giving up easily, and help students to solve problems they may face in their own lives.

Taking all of this into consideration, there are a few basic skills and approaches that students can use to help them crack just about any age-appropriate problem that you, the teacher, throw at them.  In the interest of time, we’ll introduce just four here, though plenty of others can certainly be applied where appropriate. The most common methods for solving problems that students may encounter are:

• Guess and Check;

• Draw a picture;

• Work Backward; and

• Use an Equation with a variable.

The first method on our list of math problem-solving activities for middle school is fairly self-explanatory. In a sense, it involves a bit of reverse-engineering, as the student starts with a proposed solution and works his or her way back to the beginning of the problem to see if that solution is effective.  

Drawing a picture may be more effective for visual learners, as it enables students to lay eyes on the problem and conceive of a solution in ways that they may not have otherwise.

Working backward is like a more scientific version of guessing and checking.  Students can use the information provided to step backward one piece at a time, like Guy Pearce in Memento, until they reach the solution that is in accordance with all of the details provided in the problem.

Finally, an equation that uses a variable can be effective when information is missing, or when an approach unlike the first three is required.

Again–these approaches are mere suggestions that students can apply to solving problems that they may encounter.  Ultimately, a healthy combination of different tactics can serve a student well in handling any problem thrown their way.  Skills such as these, though tough to develop at first, can go a long way toward helping US students stand up to their peers around the world in global math benchmarks, while also making day-to-day problems that they face easier to solve.  

Without further ado, here are ten math problem-solving activities for middle school students that can help them develop a number of crucial skills.  If you find these interesting, you may also like our article on the Best Problem Solving Activities For Middle School. Beyond just math, there are other areas where problem-solving can be extremely useful for that age group.  And now, the problems.

10. Missing Mangoes

Starting with the problem number ten on our list of math problem-solving activities for middle school – one night a hungry King couldn’t sleep.  He went down to the kitchen, where he found a crate full of mangoes. To satisfy his hunger, he took 1/6 of the mangoes.  Later that same night, the Queen came down and found the same crate of mangoes.  She took 1/5 of what the King had left and went back to bed satisfied.  Still later, the Prince woke up, went to the kitchen, and ate one-fourth of the remaining mangoes. His brother, then went to the kitchen and ate 1/3 of what was then left.  Finally, the princess ate 1/2 of what remained, leaving only three mangoes for the servants.  How many mangoes were originally in the bowl?

There are a lot of different ways to solve this problem, including the strategies listed above.  The first time I saw this problem, my natural instinct was to work backward from three until I reached the answer of 18.  Contrarily, you could, in theory, guess any number that you thought might work and check piecemeal to see if you ended up with 3 at the end of your permutations.

Visual learners might draw a picture of a rectangle representing all of the mangoes and subsequently draw lines to divide the rectangle into equal pieces as each member of the royal family took their share.  What you would find when all is said and done is that there are six equal bars.  If the final bar is 3, then the other 5 bars must also be 3, as they are equivalent to the final bar, meaning that in total there were 6 x 3= 18 mangoes, to begin with.

9. Mystery Weight

Continuing with a problem number 9 on our list of math problem-solving activities for middle school – when three girls jumped on a weighing scale together, it read 360 pounds. One girl stepped off and the scale moved down to 230 pounds.  One more girl jumped off and the scale read 120 pounds.  What was each girl’s weight?

Here, a simple working backward strategy will be effective in distinguishing each girl’s weight.

8. Wild Dog

Next up on the list of math problem-solving activities for middle school follows – over a five-day span, a wild dog ate a total of 105 nuts.  Each day, the dog devoured eight more nuts than he had on the day before.  How many nuts did the dog eat on each of the five days?

Knowing that the dog ate an additional 8 nuts each day, you can multiply 8 nuts by 5 days to get a total of 40.  105-40=65 and 65/5=13.  Your students now know that the first day’s nuts can’t be more than 13.  Subtracting 13 from the 8 nuts that the dog ate, you get 5, the amount eaten on the first day.  So, the dog ate 5 the first day, 13 the second day, 21 the third day, 29 the fourth day, and 37 the fifth day.

7. Castaways and Coconuts

At number 7 on our list of math problem-solving activities for middle school we have the following problem – three sailors were stuck on a deserted island that was also home to a hoard of monkeys. The sailors worked all day to collect coconuts but were too tired that night to count them.  Instead, they agreed to divide them equally the next day.  During the night, one of the sailors who couldn’t sleep decided to take his share of the coconuts. When he did so, he found that the coconuts could be divided into three equal piles, with one coconut left over, which he gave to the monkeys.  Having done this, he put his coconuts in a pile down the beach and left the rest in one pile near where the others were sleeping.  Later that night, sailor number two woke up and also decided to take the coconuts that belonged to him.  He was able to make three equal piles as well with one coconut remaining, which he threw to the monkeys.  A bit later, the final sailor woke up and did the same thing as the previous two.  In the morning, all three sailors noticed that the pile was much smaller, but each thought that he knew why and said nothing. When they then split up what remained of the original pile of coconuts equally, each sailor received seven and one was left over, which they threw to the monkeys.  How many coconuts were in the original pile?

Just as in the mango problem, students can work backward from the initial amount of 7 each by multiplying by 3 and adding 1 to wind up with 22.  From there, following the information provided in the problem will ultimately yield an answer of 79.

6. Full HD

When stores list the size of TVs in inches, they are referring to the diagonal measurement of the tv.  Comparing prices of different sized TVs and using the Pythagorean theorem, students can work to figure out how much an inch is worth for a tv in different stores.  This activity number 6 on our list of math problem-solving activities for middle school can also lead to a discussion with students discussing how much they think an extra inch should cost and how much they would be willing to pay.

5. Family Matters

How many ancestors do you have as you go back in time? Students use exponential growth to see how many people they’re related to throughout different generations of human history.  This exercise can be broken up in rounds (i.e. how many people from your parents’ generation are you related to; how many people from your grandparents’ generation; etc.). One of the most interesting math problem solving activities for middle school, don’t you think? Let’s see what else we have on our list of math problem-solving activities for middle school.

 

4. Cookie Decorations

Nicole is baking 100 cookies for the school bake sale.  While decorating, she gets bored and decides to vary the decorations.  She places a cherry on every third cookie, the icing on every fourth cookie, and puts a nut on every fifth cookie.  How many of the 100 cookies will have all three toppings on them?  Guess and check here will take some time, but would certainly be effective–after a while.  A simpler strategy would be to apply an algorithm.  1 in 3 cookies would be 33/100; 1 in 4 cookies would be 25.  1 in 5 cookies would be 20.  Finding the greatest common factor of 3, 4 and 5 will yield the students’ result (60).

 

3. Show Me the Money!

Here comes the problem number 3 on our list of math problem-solving activities for middle school – have you ever wondered how much different professions earn in a year? Students use proportional reasoning and ratios to compare how much Tom Brady and teachers make.  Bonus exercise: how much do different professions pay in taxes?

 

2. Domino Effect

How much does Domino’s charge for pizza? Students use linear functions — slope, y-intercept, and equations — to explore how much the famous pizzas really cost. Bonus, students can explore discounting.  Often, pizza companies will charge less per pizza the more pizzas you order. Using linear equations and ratios, students can figure out what percent discount an extra pizza is worth. And now, let’s see the number one on our list of math problem-solving activities for middle school.

 

1. Supersize Me!

How long would it take to burn off the calories from the food on McDonald’s menu? Students can examine McDonald’s (or other fast food restaurant menu health stats) as well as proportional reasoning and unit rates to figure out how long they’d have to exercise to burn off different McDonald’s menu items.  This activity can be further broken down by the type of exercise that a student does (ex. How long do I have to run vs. walk vs. bike vs. swim?).

We hope that you’ve found our list of math problem-solving activities for middle school useful!