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This can vary in difficulty depending on how complex you make it.

I always liked those puzzles that ask you how many rectangles or squares are in the image, and proceed to have a complex design of overlapping squares that make up more squares. Sometimes it's cool to see how students think and how they plan on counting all of the squares in the image.

Students could work individually or in groups, online or together at a whiteboard. Lots of options.

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Rectangles with Whole Area and Fractional Sides

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Rectangles with Whole Area and Fractional Sides

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This can be used in grade 9 or 10 classroom.

Can you find a rectangle that has a whole number for its area, but has non-whole numbers for side lengths?

We can start by introducing some simple rectangles to show that when the side length are whole numbers, the area is also going to be a whole number.

You can use algebra tiles and count up all of the squares that would fit in a rectangle of your choosing!

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This could be used in a grade 9 or 10 classroom.

Problem: Which is larger, n+10, or 2n+3?

Students work in groups to write a solution. Methods may vary and include guess-and-check, tables of values, graphing, and using algebra. Once students have come to a decision, they write their solution on chart paper and present to the class. Class discussion follows on different methods used and the connections between them. Students offer feedback to their peers to refine solutions.

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You may wish to explore the problem Which Is Cheaper? before working on this task.

Which is bigger, $n+10$, or $2n+3$?
How did you decide?

Here's how I decided:

"I wonder what happens when $n=4$."

"$4+10=14$ but $2 \times 4 + 3$ is only $11$."

"So it looks like $n+10$ is bigger."

But then my friend said:

"I wonder what happens when $n=10$."

"$10+10=20$ but $2 \times 10 +3$ is $23$."

"So it looks like $2n+3$ is bigger."

Can you explain why we have come to different conclusions?
Is there a diagram you could draw that would help?

For the following pairs of expressions, can you work out when each expression is bigger?

Here are some challenges to try:

Find two expressions so that one is bigger whenever $n< 5$ and the other is bigger whenever $n> 5$.

Find three expressions so that the first is biggest whenever $n< 0$, the second is biggest whenever $n$ is between 0 and 4, and the third is biggest whenever $n> 4$.

Find three expressions so that the first is biggest whenever $n< 3$, the second is biggest when $n> 3$, and the third is never the biggest.

Find three expressions so that one of them is the biggest regardless of the value of $n.$

You may also be interested in the other problems in our Many ways to see Feature.

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You may wish to explore the problem Which Is Cheaper? before working on this task. Which is bigger, $n+10$, or $2n+3$? How did you decide? Here's how I decided: "I wonder what happens when $n=4$." "$4+10=14$ but $2 \times 4 + 3$ is only $11$." "So it looks like $n+10$ is bigger."

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This could be used for any high school math class. My AT used it as a time filler at the end of class for fun.

WITHOUT REVEALING THEIR NUMBER students follow these steps:
1. Pick any 6 digit number
2. Write that number backwards
3. Subtract the numbers (smaller from larger to get a positive number)
4. Multiply the result by any number from 1 - 9
5. Tell the teacher all but 1 of the digits in the result.

Every time the teacher could correctly guess what number was missing.

To make this into a rich task, demonstrate it at the board, then have students code the steps so that the computer takes their place (generating the number) and the user has to guess the correct digit. The code should say if you're right or not.

-Kirstin

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This would be a task suited for grade 12 calculus students.

Students in groups could be given a type of function (quadratic, exponential, logarithmic, and so on) for which they would have to create an optimization problem/question.

i.e. given the topic of exponential functions, the students could be responsible for coming up  with a problem such as: "The number of bus riders from the suburbs to downtown per day is represented by 1200(1.15)^(-x), where x is the fare in dollars. What fare will maximize the total revenue?" (from the Ontario curriculum)

This could be done as a jigsaw activity where each group creates a problem and then solves all other groups' problems. Students can evaluate one another afterwards to determine if their problems were posed effectively or if they needed to be improved. I would facilitate this refining stage by floating around and guiding the students.

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Ask students to go around their town/city and take a picture of a structure (i.e. bridge, streetlight, building). If they can't take a picture, they may use a picture from either a newspaper or through an online search engine. They will give a reasonable height to this structure and pick a "base" as to where they are looking to the structure. They will need to find the "hypotenuse" of this distance between the base and height. After they will discuss with classmates and compare their findings. This may be used as a warm-up activity in a grade nine classroom to start exploring the idea of triangles and Pythagorean Theorem. - Katie

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This could be used in grades 10-12.

The quadratic formula is often misused, students have difficulties carrying negatives, inputting and isolating variables, etc. Coding the formula, will break the formula down and demand a conceptual understanding.

Have students with the assistances of basic instructions code the quadratic formula and test it out. They may need to refine their code and/or logic, but inputting a series of test numbers, until they reach the correct solutions. Have students share their code, and test with others. Until the code works for each case, and is robust.

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This could be used in Grade 8 - Grade 10 classroom.

When you add a number (i.e., 24) to its reverse (42) you often get a palindrome (66). Does this always work?

Conjecture: Every number becomes a palindrome after some number of steps

Alone or in pairs, students are given a 100s-chart to record data. They can record by colour. For example, if they want to check the number 18. I would add 18 to its reverse, 81, and get 99. That’s a palindrome, and that took 1 step. So I can colour 18 blue. They then test 19, which adds to 91 to make 110. They then add 110 to its reverse (011) to get 121 - that's a palindrome! They can colour code this as red, because it took 2 steps.

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This can be used in a grade 8/9 classroom.

Follow the instructions:
Pick a whole number between 1 and 10. Add 2. Multiply by 2. Subtract 2. Divide by 2. Subtract your original number

Questions to follow:
Will this work with fractions?
Why does this always work?
What happens if we double each line in the trick?
Can we generalize?

It offers low floor-high ceiling because students can use it to practice arithmetic or dig deeper with the questions presented and look at the bigger picture.

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Could use in Grade 8 - Grade 12 classroom.

Decompose 25 using addition. For example:
25 = 10 + 15
25 = 10 + 10 + 5
What is the biggest product you can make if you multiply the addends together?

In groups, students are responsible for problem solving and thinking together as a team as they look for patterns and relationships to solve the problem.

Have students work at the whiteboards/chalkboards in the classroom. Once the task is done, consolidate findings through discussion. Have students summarize the findings/concepts in their notes.