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Module 1: Problem Solving Strategies

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Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

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Logical Puzzles

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• Andrew Hayes

A logical puzzle is a problem that can be solved through deductive reasoning. This page gives a summary of the types of logical puzzles one might come across and the problem-solving techniques used to solve them.

Elimination Grids

Truth tellers and liars, cryptograms, arithmetic puzzles, river crossing puzzle, tour puzzles, battleship puzzles, chess puzzles, k-level thinking, other puzzles.

Main Article: Propositional Logic See Also: Predicate Logic

One of the simplest types of logical puzzles is a syllogism . In this type of puzzle, you are given a set of statements, and you are required to determine some truth from those statements. These types of puzzles can often be solved by applying principles from propositional logic and predicate logic . The following syllogism is from Charles Lutwidge Dodgson, better known under his pen name, Lewis Carroll.

I have a dish of potatoes. The following statements are true: No potatoes of mine, that are new, have been boiled. All my potatoes in this dish are fit to eat. No unboiled potatoes of mine are fit to eat. Are there any new potatoes in this dish? The first and third statements can be connected by a transitive argument. All of the new potatoes are unboiled, and unboiled potatoes aren't fit to eat, so no new potatoes are fit to eat. The second statement can be expressed as the equivalent contrapositive. All of the potatoes in the dish are fit to eat; if there is a potato that is not fit to eat, it isn't in the dish. Then, once again, a transitive argument is applied. New potatoes aren't fit to eat, and inedible potatoes aren't in the dish. Thus, there are no new potatoes in the dish. \(_\square\)

Given below are three statements followed by three conclusions. Take the three statements to be true even if they vary from commonly known facts. Read the statements and decide which conclusions follow logically from the statements.

Statements: 1. All actors are musicians. 2. No musician is a singer. 3. Some singers are dancers.

Conclusions: 1. Some actors are singers. 2. Some dancers are actors. 3. No actor is a singer.

Answer Choices: a) Only conclusion 1 follows. b) Only conclusion 2 follows. c) Only conclusion 3 follows. d) At least 2 of the conclusions follows.

Main Article: Elimination Grids

Some logical puzzles require you to determine the correct pairings for sets of objects. These puzzles can often be solved with the process of elimination, and an elimination grid is an effective tool to apply this process.

An example of an elimination grid

Elimination grids are aligned such that each row represents an object within a set, and each column represents an object to be paired with an object from that set. Check marks and X marks are used to show which objects pair, and which objects do not pair.

Mr. and Mrs. Tan have four children--three boys and a girl-- who each like one of the colors--blue, green, red, yellow-- and one of the letters--P, Q, R, S.

• The oldest child likes the letter Q.
• The youngest child likes green.
• Alfred likes the letter S.
• Brenda has an older brother who likes R.
• The one who likes blue isn't the oldest.
• The one who likes red likes the letter P.
• Charles likes yellow.

Based on the above facts, Darius is the \(\text{__________}.\)

Main Article: Truth-Tellers and Liars

A variation on elimination puzzles is a truth-teller and liar puzzle , also known as a knights and knaves puzzle . In this type of puzzle, you are given a set of people and their respective statements, and you are also told that some of the people always tell the truth and some always lie. The goal of the puzzle is to deduce the truth from the given statements.

20\(^\text{th}\) century mathematician Raymond Smullyan popularized these types of puzzles.

You are in a room with three chests. You know at least one has treasure, and if a chest has no treasure, it contains deadly poison.

Each chest has a message on it, but all the messages are lying .

• Left chest: "The middle chest has treasure."
• Middle chest: "All these chests have treasure."
• Right chest: "Only one of these chests has treasure."

Which chests have treasure?

There are two people, A and B , each of whom is either a knight or a knave.

A says, "At least one of us is a knave."

What are A and B ?

\(\) Details and Assumptions:

• A knight always tells the truth.
• A knave always lies.

Main Article: Cryptograms

A cryptogram is a puzzle in which numerical digits in a number sentence are replaced with characters, and the goal of the puzzle is to determine the values of these characters.

\[ \begin{array} { l l l l l } & &P & P & Q \\ & &P & Q & Q \\ + && Q & Q & Q \\ \hline & & 8 & 7 & 6 \\ \end{array} \]

In the sum shown above, \(P\) and \(Q\) each represent a digit. What is the value of \(P+Q\)?

\[ \overline{EVE} \div \overline{DID} = 0. \overline{TALKTALKTALKTALK\ldots} \]

Given that \(E,V,D,I,T,A,L\) and \(K \) are distinct single digits, let \(\overline{EVE} \) and \( \overline{DID} \) be two coprime 3-digit positive integers and \(\overline{TALK} \) be a 4-digit integer, such that the equation above holds true, where the right hand side is a repeating decimal number.

Find the value of the sum \( \overline{EVE} + \overline{DID} + \overline{TALK} \).

Main Articles: Fill in the Blanks and Operator Search

Arithmetic puzzles contain a series of numbers, operations, and blanks in order, and the object of the puzzle is to fill in the blanks to obtain a desired result.

\[\huge{\Box \times \Box \Box = \Box \Box \Box}\]

Fill the boxes above with the digits \(1,2,3,4,5,6\), with no digit repeated, such that the equation is true.

Enter your answer by concatenating all digits in the order they appear. For example, if the answer is \(1 \times 23 = 456\), enter \(123456\) as your final answer.

Also try its sister problem.

\[ \LARGE{\begin{eqnarray} \boxed{\phantom0} \; + \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; - \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \times \; \boxed{\phantom0}\; &=& \; \boxed{\phantom0} \\ \boxed{\phantom0} \; \div \; \boxed{\phantom0} \; &=& \; \boxed{\phantom0} \\ \end{eqnarray}} \]

Put one of the integers \(1, 2, \ldots , 13\) into each of the boxes, such that twelve of these numbers are used once for each (and one number is not used at all) and all four equations are true.

What is the sum of all possible values of the missing (not used) number?

Main Article: River Crossing Puzzles

In a river crossing puzzle , the goal is to find a way to move a group of people or objects across a river (or some other kind of obstacle), and to do it in the fewest amount of steps or least amount of time.

A famous river crossing problem is Richard Hovasse's bridge and torch problem , written below.

Four people come to a river in the night. There is a narrow bridge, but it can only hold two people at a time. They have one torch and, because it's night, the torch has to be used when crossing the bridge. Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes. When two people cross the bridge together, they must move at the slower person's pace. The question is, can they all get across the bridge in 15 minutes or less? Assume that a solution minimizes the total number of crosses. This gives a total of five crosses--three pair crosses and two solo crosses. Also, assume we always choose the fastest for the solo cross. First, we show that if the two slowest persons (C and D) cross separately, they accumulate a total crossing time of 15. This is done by taking persons A, C, D: D+A+C+A = 8+1+5+1=15. (Here we use A because we know that using A to cross both C and D separately is the most efficient.) But, the time has elapsed and persons A and B are still on the starting side of the bridge and must cross. So it is not possible for the two slowest (C and D) to cross separately. Second, we show that in order for C and D to cross together that they need to cross on the second pair cross: i.e. not C or D, so A and B, must cross together first. Remember our assumption at the beginning states that we should minimize crosses, so we have five crosses--3 pair crossings and 2 single crossings. Assume that C and D cross first. But then C or D must cross back to bring the torch to the other side, so whoever solo-crossed must cross again. Hence, they will cross separately. Also, it is impossible for them to cross together last, since this implies that one of them must have crossed previously, otherwise there would be three persons total on the start side. So, since there are only three choices for the pair crossings and C and D cannot cross first or last, they must cross together on the second, or middle, pair crossing. Putting all this together, A and B must cross first, since we know C and D cannot and we are minimizing crossings. Then, A must cross next, since we assume we should choose the fastest to make the solo cross. Then we are at the second, or middle, pair crossing, so C and D must go. Then we choose to send the fastest back, which is B. A and B are now on the start side and must cross for the last pair crossing. This gives us, B+A+D+B+B = 2+1+8+2+2 = 15. It is possible for all four people to cross in 15 minutes. \(_\square\)

Main Article: Tour Puzzles See Also: Eulerian Path

In a tour puzzle , the goal is to determine the correct path for an object to traverse a graph. These kinds of puzzles can take several forms: chess tours, maze traversals, eulerian paths , and others.

Find the path that leads from the star in the center back to the star in the center. Paths can only go in the direction of an arrow. Image Credit: Eric Fisk Show Solution The solution path is outlined in red below.

Determine a path through the below graph such that each edge is traversed exactly once . Show Solution There are several possible solutions. One possible solution is shown below, with the edges marked in the order they are traversed.

A chess tour is an interesting type of puzzle in its own right, and is explained in detail further down the page.

Main Article: Nonograms

A nonogram is a grid-based puzzle in which a series of numerical clues are given beside a rectangular grid. When the puzzle is completed, a picture is formed in the grid.

The puzzle begins with a series of numbers on the left and above the grid. Each of these numbers represents a consecutive run of shaded spaces in the corresponding row or column. Each consecutive run is separated from other runs by at least one empty space. The puzzle is complete when all of the numbers have been satisfied. The primary technique to solve these puzzles is the process of elimination. If the puzzle is designed correctly, there should be no guesswork required.

Complete the nonogram: Show Solution

One of the many logical puzzles is the Battleship puzzle (sometimes called Bimaru, Yubotu, Solitaire Battleships or Battleship Solitaire). The puzzle is based on the Battleship game.

Solitaire Battleships was invented by Jaime Poniachik in Argentina and was first featured in the magazine Humor & Juegos.

This is an example of a solved Battleship puzzle. The puzzle consists of a 10 × 10 small squares, which contain the following:

• 1 battleship 4 squares long
• 2 cruisers 3 squares long each
• 3 destroyers 2 squares long each
• 4 submarines 1 square long each.

They can be put horizontally or vertically, but never diagonally. The boats are placed so that no boats touch each other, not even vertically. The numbers beside the row/column indicate the numbers of squares occupied in the row/column, respectively. ⬤ indicates a submarine and ⬛ indicates the body of a ship, while the half circles indicate the beginning/end of a ship.

The goal of the game is to fill in the grid with water or ships.

Main Article: Sudoku

A sudoku is a puzzle on a \(9\times 9\) grid in which each row, column, and smaller square portion contains each of the digits 1 through 9, each no more than once. Each puzzle begins with some of the spaces on the grid filled in. The goal is to fill in the remaining spaces on the puzzle. The puzzle is solved primarily through the process of elimination. No guesswork should be required to solve, and there should be only one solution for any given puzzle.

Solve the sudoku puzzle: Puzzle generated by Open Sky Sudoku Generator Each row should contain the each of the digits 1 through 9 exactly once. The same is true for columns and the smaller \(3\times 3\) squares. Show Solution

Main Article: Chess Puzzles See Also: Reduced Games , Opening Strategies , and Rook Strategies

Chess puzzles take the rules of chess and challenge you to perform certain actions or deduce board states.

One kind of chess puzzle is a chess tour , related to the tour puzzles mentioned in the section above. This kind of puzzle challenges you to develop a tour of a chess piece around the board, applying the rules of how that piece moves.

Dan and Sam play a game on a \(5\times3\) board. Dan places a White Knight on a corner and Sam places a Black Knight on the nearest corner. Each one moves his Knight in his turn to squares that have not been already visited by any of the Knights at any moment of the match.

For example, Dan moves, then Sam, and Dan wants to go to Black Knight's initial square, but he can't, because this square has been occupied earlier.

When someone can't move, he loses. If Dan begins, who will win, assuming both players play optimally?

This is the seventeenth problem of the set Winning Strategies.

Due to its well-defined ruleset, the game of chess affords many different types of puzzles. The problem below shows that you can even deduce whose turn it is from a certain boardstate (or perhaps you cannot).

Whose move is it now?

Main Article: K-Level Thinking See Also: Induction - Introduction

K-level thinking is the name of a kind of assumption in certain logic puzzles. In these types of puzzles, there are a number of actors in a situation, and each of them is perfectly logical in their decision-making. Furthermore, each of these actors is aware that all other actors in the situation are perfectly logical in their decision-making.

Calvin, Zandra, and Eli are students in Mr. Silverman's math class. Mr. Silverman hands each of them a sealed envelope with a number written inside.

He tells them that they each have a positive integer and the sum of the three numbers is 14. They each open their envelope and inspect their own number without seeing the other numbers.

Calvin says,"I know that Zandra and Eli each have a different number." Zandra replies, "I already knew that all three of our numbers were different." After a brief pause Eli finally says, "Ah, now I know what number everyone has!"

What number did each student get?

Format your answer by writing Calvin's number first, then Zandra's number, and finally Eli's number. For example, if Calvin has 8, Zandra has 12, and Eli has 8, the answer would be 8128.

Two logicians must find two distinct integers \(A\) and \(B\) such that they are both between 2 and 100 inclusive, and \(A\) divides \(B\). The first logician knows the sum \( A + B \) and the second logician knows the difference \(B-A\).

Then the following discussion takes place:

Logician 1: I don't know them. Logician 2: I already knew that.

Logician 1: I already know that you are supposed to know that. Logician 2: I think that... I know... that you were about to say that!

Logician 1: I still can't figure out what the two numbers are. Logician 2: Oops! My bad... my previous conclusion was unwarranted. I didn't know that yet!

What are the two numbers?

Enter your answer as a decimal number \(A.B\). \((\)For example, if \(A=23\) and \(B=92\), write \(23.92.)\)

Note: In this problem, the participants are not in a contest on who finds numbers first. If one of them has sufficient information to determine the numbers, he may keep this quiet. Therefore nothing may be inferred from silence. The only information to be used are the explicit declarations in the dialogue.

Of course, the puzzles outlined above aren't the only types of puzzles one might encounter. Below are a few more logical puzzles that are unrelated to the types outlined above.

You are asked to guess an integer between \(1\) and \(N\) inclusive.

Each time you make a guess, you are told either

(a) you are too high, (b) you are too low, or (c) you got it!

You are allowed to guess too high twice and too low twice, but if you have a \(3^\text{rd}\) guess that is too high or a \(3^\text{rd}\) guess that is too low, you are out.

What is the maximum \(N\) for which you are guaranteed to accomplish this?

\(\) Clarification : For example, if you were allowed to guess too high once and too low once, you could guarantee to guess the right answer if \(N=5\), but not for \(N>5\). So, in this case, the answer would be \(5\).

You play a game with a pile of \(N\) gold coins.

You and a friend take turns removing 1, 3, or 6 coins from the pile. The winner is the one who takes the last coin.

For the person that goes first, how many winning strategies are there for \(N < 1000?\)

\(\) Clarification: For \(1 \leq N \leq 999\), for how many values of \(N\) can the first player develop a winning strategy?

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Mathematical Treasure Hunt - Various

Subject: Mathematics

Age range: 7-11

Resource type: Game/puzzle/quiz

Last updated

22 February 2018

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Fun teaching resources & tips to help you teach math with confidence

Math Scavenger Hunts: Simple Math Practice for the Classroom {with FREE Hunts!}

Could you use a simple classroom strategy that gets kids moving while also practicing and reviewing math skills? Try incorporating task card math scavenger hunts into your classroom and get more kids engaged in learning!

I am one of the lucky ones to have an eighty minute math block EVERY. SINGLE. DAY. But I quickly learned that my students could not spend those eighty minutes sitting in their seats . They needed to get up and MOVE , but I also still wanted them to engage in the math. Task Card Math Scavenger Hunts quickly solved this problem. My students love walking into my classroom and seeing the twenty-four task cards hung up around the room. They know that after our mini-lesson, they are in for a treat!

*This is a guest post from Paige at The Math Matrix

How to Use a Math Scavenger Hunts in the Classroom:

Our task card scavenger hunts are a pack of twenty-four task cards that require students to solve problems while “hunting down” their next problem. All students start at a different card – this way, students can complete the activity without everyone being crowded around one single problem.

At each card, students will solve the math problem . After they record their answers on the recording sheet (in the correct box), they need to search the room for the solution , which brings them to the next card with a new problem to solve.

Your students will be going around the room to continue to solve and search for the next answer until they get back to the card they started at – this will ensure they got all of the answers correct!

If they end up at their starting card and yet have missed some problems, they know they’ve made a mistake somewhere.

Or if they can’t find their solution around the room, perhaps it’s because they don’t have the right solution!

Using the scavenger hunt setup, students can self-check and work together to correct mistakes.

Four Benefits of Using Math Task Card Scavenger Hunts:

1. Get Students TALKING

I love when my students talk math in my classroom, and encourage discussions about solution strategies and helping each other with incorrect answers.

2. Get Students MOVING

I hang up my Task Card Scavenger Hunts all over my room and grade level hallway and have my students use their clipboards to get up and MOVE around!

3. Get Students IMMEDIATE FEEDBACK

The nature of Task Card Scavenger Hunts gives immediate feedback because if the student cannot find the answer on another card, they know that they made a mistake.

While introducing this activity, I always make sure to discuss what might happen if we can’t find the answer and to circle back to the card they were working on to troubleshoot their solution.

4. Get Students HELP

When my students are working all over my room, I find it really frees up my time and I can move from student to student to give help to those who need it. I also found that this time is the best to pull small groups of students when I see them making the same mistakes or having the same question!

Trying Task Card Scavenger Hunts in Your Classroom

As you can tell, Task Card Scavenger Hunts really help my students break up their time in my class, while still practicing the important math concepts that they have been learning.

If you are looking for a free sample to see if math scavenger hunts are right for your classroom, CLICK HERE for a free sample covering mixed multiplication and division math facts !

Plus, there are many more task card scavenger hunts over in my store for you to check out!

Once you start with Task Card Scavenger Hunts, you won’t look back ( I promise – your students will be BEGGING to do more! )

Find More Math Scavenger Hunts for the Classroom Below:

• Christmas Multiplication Facts Scavenger Hunt – Grades 3-4 {FREE}
• 2-3 Digit Multiplication Christmas Scavenger Hunt – 4th Grade {FREE}
• Area & Perimeter Practice for 3rd Grade – Scavenger Hunt Activity {FREE}
• Virtual Math Scavenger Hunt: Learn About Math in Nature {FREE}

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Join 165,000+ parents and teachers who learn new tips and strategies, as well as receive engaging resources to make math fun. Plus, receive my guide, "5 Games You Can Play Today to Make Math Fun," as my free gift to get you started!

30 Math Riddles (With Answers)

Math may not be everyone’s cup of tea, but when you think about it, it’s everywhere. Most of us likely do some type of math just about every day. So how can we make learning math and improving our problem-solving abilities more enjoyable? The answer is easy—riddles! And these math riddles are a great place to start.

These riddles span various age levels and abilities. Try some out yourself or gather the kids and make it a family or school activity. Math is fundamental, so there’s no time like the present to fine-tune those skills!

Calling All Math Masters!

If you enjoy math riddles, you’re bound to love the Math Masters category of our in-home scavenger hunts ! All you’ll need is your family, your internet-enabled device, and your competitive spirit. Our intuitive interface will have you connected, solving clues, and completing challenges in no time.

Developed in conjunction with 2 elementary school teachers, this hunt makes problem-solving a blast! When you’re done with that category, there are several more from which you can choose. The one thing they all have in common is FUN!

1. Riddle: A merchant can place 8 large boxes or 10 small boxes into each carton for shipping. In one shipment, he sent a total of 96 boxes. If there are more large boxes than small boxes, how many cartons did he ship? Answer: 11 cartons total: 7 cartons of large boxes (7 * 8 = 56 boxes), 4 cartons of small boxes (4 * 10 = 40 boxes).

2. Riddle: I have a calculator that can display ten digits. How many different ten-digit numbers can I type using just the 0-9 keys once each, and moving from one keypress to the next using the knight’s move in chess? (In chess, the knight move in an L-shape—one square up and two across, two squares down and one across, two squares up and one across, and other like combinations.) Answer: You can form the numbers 5034927618 and 5038167294. You can also form their reverses: 8167294305 and 4927618305. Hence four different numbers can be made.

3. Riddle: When my dad was 31, I was just 8 years old. Now his age is twice as old as my age. What is my present age? Answer: When you calculate the difference between the ages, you can see that it is 23 years. So you must be 23 years old now.

4. Riddle: An insurance salesman walks up to a house and knocks on the door. A woman answers, and he asks her how many children she has and how old they are. She says I will give you a hint. If you multiply the 3 children’s ages, you get 36. He says this is not enough information. So she gives a him 2nd hint. If you add up the children’s ages, the sum is the number on the house next door. He goes next door and looks at the house number (13) and says this is still not enough information. So she says she’ll give him one last hint which is that her oldest of the 3 plays piano. Why would he need to go back to get the last hint after seeing the number on the house next door? Answer: After the first hint, there are several different options. When he realizes the sum is 13, the answer is one of two possibilities. The third hint indicates there is an “oldest” daughter. Since one of the two options includes older twins (1 + 6 + 6 = 13) and the other includes young twins (2 + 2 + 9 = 13), the daughters must be 2, 2, and 9.

5. Riddle: Scott has $28.75. He purchased three cookies that cost $1.50 each, five newspapers that each cost $0.50, five flowers for $1.25 each, and used the remainder of the cash on a pair of sunglasses. How much were the sunglasses? Answer: $15.50.

6. Riddle: What is the smallest whole number that is equal to seven times the sum of its digits? Answer: 21. The two-digit number ab stands for 10a + b since the first digit represents 10s and the second represents units. If 10a + b = 7(a + b), then 10a + b = 7a + 7b, and so 3a = 6b, or, more simply, a = 2b. That is, the second digit must be twice the first. The smallest possible number is 21.

7. Riddle: A monkey is trying to climb a coconut tree. He takes 3 steps forward and slips back 2 steps downward. Each forward step is 30 cm and each backward step is 40 cm. How many steps are required to climb a 100 cm tree? Answer: 50 steps.

8. Riddle: In a mythical land 1/2 of 5 = 3. If the same proportion holds, what is the value of 1/3 of 10? Answer: 4.

9. Riddle: It is 9 am now. Rita studies for 2 hours, takes a bath for 1 hour, and then has lunch for 1 hour. How many hours are left before 9 am tomorrow? Answer: 20 hours.

10. Riddle: There is an empty basket that is one foot in diameter. What is the total number of eggs that you can put in this empty basket? Answer: Only one egg! Once you put an egg into the basket, it’s no longer an empty basket!

Need a break from all that math? Our list of funny riddles is giggle-worthy!

11. Riddle: You know 2 + 2 gives you the same total as 2 x 2. Find a set of three different whole numbers whose sum is equal to their total when multiplied. Answer: The three different whole numbers whose sum is equal to their total when multiplied are 1, 2, and 3.

12. Riddle: If you multiply this number by any other number, the answer will always be the same. What is the number? Answer: Zero.

13. Riddle: A grandfather, two fathers, and two sons went to a movie theater together and everyone bought one movie ticket each. Movie tickets cost $7.00 and their total was $21.00. How? Answer: They purchased 3 tickets. The grandfather is also a father and the father is also a son.

14. Riddle: A small number of cards has been lost from a complete deck. If I deal cards to four people, three cards remain. If I deal to three people, two remain. If I deal to five people, two cards remain. How many cards are there? Answer: There are 47 cards. A complete deck is 52 cards. 5 cards have been lost.

15. Riddle: Tom was on the way to the park. He met a man with seven wives and each of them came with seven sacks. All these sacks contain seven cats and each of these seven cats had seven kittens. So in total, how many were going to the park? Answer: 1. Only Tom was going to the park.

16. Riddle: I am an odd number; take away a letter and I become even. What number am I? Answer: Seven. (SEVEN-S=EVEN.)

17. Riddle: How can you add eight 8’s to get the number 1,000? Answer: 888 + 88 + 8 + 8 + 8 = 1,000.

18. Riddle: If there are four apples and you take away three, how many do you have? Answer: Three apples.

19. Riddle: I am one with a couple of friends. Quarter a dozen, and you’ll find me again. What number am I? Answer: 3.

20. Riddle: Tom weighs half as much as Peter and Jerry weigh three times the weight of Tom. Their total weight is 720 pounds. Can you figure out the individual weights of each man? Answer: Tom weighs 120, Peter weighs 240, and Jerry weighs 360.

Find these math riddles too hard? How about some easy riddles ?

21. Riddle: If it is two hours later, then it will take half as much time till it’s midnight as it would be if it were an hour later. What time is it? Answer: 9:00 PM.

22. Riddle: In an alien land far away, half of 10 is 6. If the same proportion holds true, then what is 1/6th of 30 in this alien land? Answer: 6.

23. Riddle: In two years, Tom will be twice as old as he was five years ago. How old is Tom? Answer: 12.

24. Riddle: A little boy goes shopping and purchases 12 tomatoes. On the way home, all but 9 get mushed and ruined. How many tomatoes are left in good condition? Answer: 9 tomatoes

25. Riddle: A carton contains apples that were divided into two equal parts and sold to two traders Tarun and Tanmay. Tarun had two fruit shops and decided to sell an equal number of apples in both shops, A and B respectively. A mother visited shop A and bought all the apples in the shop for her kids. But one apple was left after dividing all the apples among her children. If each child got one apple, what is the minimum number of apples in the carton? Answer: 12. That is the minimum possible number of apples for each child to have at least one.

26. Riddle: A duck was given $9, a spider was given $36, and a bee was given $27. Based on this information, how much money would be given to a cat? Answer: $18 ($4.50 per leg).

27. Riddle: The speed of a train is 3 meters/second and it takes 10 seconds to cross a lamp post. What is the length of the train? Answer: 30 meters. The time taken by the train to pass the stationary object is equal to the length of the train divided by the speed of the train.

28. Riddle: It takes 12 men 12 hours to construct a wall. Then how long will it take for 6 men to complete the same wall? Answer: It would probably take 24 hours, but there is no need to make it again. The job is already done!

29. Riddle: What are four consecutive prime numbers that, when added, equal 220? Answer: 220= 47+ 53+ 59+ 61

30. Riddle: An athlete can jump FOREVER. However, every time she jumps, she goes half as far as her prior jump. On her very first jump, she goes half a foot. On her second jump, she goes a quarter of a foot, and so. How long will it take her to get a foot away from her starting point? Answer: She will NEVER travel a full foot because the distance keeps being reduced by half.

Say it with us—math is cool! Which of our math questions got you thinking? Share your thoughts in the comments section.

And don’t forget to try a Let’s Roam Scavenger Hunt or any of our other activities that will let you roam from home . When you’re ready to get out and about, we have city art walks and scavenger hunts all over the world. Try one today!

Frequently Asked Questions

Use puzzles, riddles, and games based on mathematics. This list of math riddles should help! Or consider a math-themed scavenger hunt —you’ll have a little math master on your hands!

Math riddles make math more fun and, when learning is fun, it becomes easier. Before you know it, you (or your children) will be thinking like a true mathematician!

888 + 88 + 8 + 8 + 8 = 1,000. Find this and 30 math riddles at LetsRoam.com, along with other family-friendly activities , including a math-themed scavenger hunt .

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Mathematical Reasoning & Problem Solving

In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.

Previously Covered:

• Over the course of the previous lesson, we reviewed some basics about chance and probability, as well as some basics about sampling, surveys, etc. We also covered some ideas about data sets, how they’re represented, and how to interpret the results.

Approaches to Problem Solving

When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.

To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:

Identify the following four-digit number when presented with the following information:

• One of the four digits is a 1.
• The digit in the hundreds place is three times the digit in the thousands place.
• The digit in the ones place is four times the digit in the ten’s place.
• The sum of all four digits is 13.
• The digit 2 is in the thousands place.

Help your students identify and prioritize the information presented.

In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.

We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.

Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.

So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:

If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.

Recall that the clues’ relevance were identified and prioritized as follows:

• clue #3 and clue #1

By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.

Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:

Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?

Let’s solve this problem by representing it in a visual way , in this case, a diagram:

You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:

b = the base of triangle EFA

h = the height of triangle EFA and the height above the ground at which the ropes intersect

If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.

Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.

Let’s try another one.

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.

How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.

Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:

How many more faces does a cube have than a square pyramid?

Reveal Answer

The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:

From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.

Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:

• Sort and prioritize relevant and irrelevant information.
• Represent a problem in different ways, such as words, symbols, concrete models, and diagrams.
• Generate and use estimations to find solutions to mathematical problems.

Mathematical Mistakes

Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.

Circular Arguments

These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:

• Premise: Only an untrustworthy man would become an insurance salesman; the fact that insurance salesmen cannot be trusted is proof of this.
• Conclusion: Therefore, insurance salesmen cannot be trusted.

While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.

Assuming the Truth of the Converse

Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”

Assuming the Truth of the Inverse

Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:

If you grew up in Minnesota , you’ve seen snow.

Now, notice that the inverse of this statement is not necessarily true:

If you didn’t grow up in Minnesota , you’ve never seen snow.

Faulty Generalizations

This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.

Faulty Analogies

It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:

People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.

False (or tenuous) analogies are often used in persuasive arguments.

Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.

Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .

Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:

If p, then q

An arrow is used to indicate that q is derived from p, like this:

This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.

A direct proof will attempt to lay out the shortest number of steps between p and q.

The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.

Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.

Given: Triangle ABC is isosceles with B marking the vertex

Prove: Angles A and C are congruent.

Now, let’s work through this, matching our statements with our reasons.

• Triangle ABC is isosceles . . . . . . . . . . . . Given
• Angle A is the vertex . . . . . . . . . . . . . . . . Given
• Angles A and C are not congruent . . Indirect proof assumption
• Line AB is equal to line BC . . . . . . . . . . . Legs of an isosceles triangle are congruent
• Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are congruent
• Contradiction . . . . . . . . . . . . . . . . . . . . . . Angles can’t be congruent and incongruent
• Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong
• Therefore, if angles A and C are not incongruent, they are congruent.

“Always, Sometimes, and Never”

Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.

Example: x < x 2 for all real numbers x

We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:

Example: For all primes x ≥ 3, x is odd.

This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.

• Know and be able to identify common mathematical errors, such as circular arguments, assuming the truth of the converse, assuming the truth of the inverse, making faulty generalizations, and faulty use of analogical reasoning.
• Be familiar with direct proofs and indirect proofs (proof by contradiction).
• Be able to work with problems to identify “always,” “sometimes,” and “never” statements.

Math in Nature Scavenger Hunt

Math doesn’t just exist in the classroom. This summer, learn all about math in nature with these free printable math activities. 

In This Article

Learn all about math in nature through photographs and interactive summer activities! We have four fun, printable math activities your kids can do alone or with light supervision. 

Jump to any of the activities below:

• Tree Ring Circus
• Fill in the Flying V

Symmetry in Nature

• The Ultimate Math in Nature Scavenger Hunt

Table of contents

Shapes in Nature

• Ultimate Math Scavenger Hunt

It’s easy when you’re sitting in class to think that math is just something you study in school. But there’s a lot more to math than adding, subtracting, and solving word problems. 

Look around and you’ll see plenty of examples of math in nature, from fun geometric shapes called fractals to the Fibonacci sequence and more. 

Let’s explore where you can find math in nature this summer — then get outside to see math in the wild!

Take a look around, and you’ll find that many of the shapes you discuss in class occur naturally in nature, too. In fact, many famous philosophers like Pythagoros probably saw certain patterns in nature and began to ponder why this pattern existed and whether it could be copied—or even just understood—by humankind! 

Let’s explore the various shapes we see in nature and what math can teach us about them.

Circles are one of the most common shapes in nature. From eyeballs to water droplets to the center of a sunflower, you can see circles almost everywhere you go. 

This makes sense, because many people believe that the circle is “the perfect shape.” That’s because most humans are attracted to curved lines. In fact, scientists say that most of us find circles calming, peaceful, and relaxing.

Give this a try...

Can you think of all the circles you notice in one day? What about the circles you might  not  notice until you start looking for them? See how many circles you can find on a summer nature walk!

We’ve talked about the circles that occur in nature. One of the most popular – and fascinating – examples of naturally occurring circles are tree rings. Not only do tree rings tell us how old a tree is, but they can also explain what the weather was like in a year, giving us insight into climate patterns. 

Use the summertime to take a walk through the woods. You’re likely to find tree ring patterns on stumps or toppled trees. Next time you find one, try to count all the rings to see how old that tree was.

Math in Nature Activity 1: Tree Ring Circus

Want some practice before you take to the woods? Download our activity “Tree Ring Circus” and discover how old these trees are!

You probably wouldn’t think that a hexagon , with its six sides, would be a popular shape in nature, but it’s one of the most common shapes to find in the wild. 

In fact, it makes sense, as hexagons are shapes that, when connected, leave the least amount of wasted space in any particular area. 

But you’re probably trying to think of places where you’ve seen hexagons and struggling to come up with examples. 

You’ll see plenty of bees around in the summertime, right? Well, bees are insects, and almost all insect eyes are made up of hexagonal patterns. However, beehives and honeycomb also contain hexagonal shapes. 

Winter might be a few months away, but keep in mind that snowflakes are hexagons, too! All snowflakes begin as small, hexagonal plates. As they fall, they collect crystals that give them their unique shape—but look at the center of any snowflake, and you’ll see a hexagon!

It’s easy to see triangles in many of the buildings and sidewalks you see every day, but did you know that there are plenty of triangles that occur out in the wild? 

Take geese migration, for example. Geese fly in a v formation whenever they are migrating from North to South and vice versa. In this formation, the lead bird acts as a windbreak, helping to conserve the energy of the birds behind it by reducing wind resistance. 

Each goose in the flock takes turns as the head of the V. 

Each bird also flies slightly above the bird in front of them, which also helps to reduce wind resistance and maintains the birds’ energy levels. 

Whenever one bird gets tired, it falls back, and the next bird takes its place. This migration pattern uses angles—and specifically, the triangle shape—to help these geese get to where they need to go!

Math in Nature Activity 2: Flying V

Let’s explore natural triangles. Download our activity “Fill in the flying v” and turn a flock of geese into colorful triangles.

Spirals and Fibonacci Sequence

The Fibonacci sequence, also known as the Golden Ratio, goes a little something like this: 1, 1, 2, 3, 5, 8, 13, 21. Basically, add the number to the number that comes before it to get the next number in the sequence, so…

And on and on forever. 

Interestingly, this pattern is incredibly popular in nature. You can find the Fibonnaci sequence in flower petals, the seed heads of sunflowers and daisies, and in pinecones. You can also find the Fibonacci sequence in conches and seashells. 

What do all of these shapes have in common? They form spirals. And they aren’t the only examples of naturally occurring spirals that follow the Fibonnaci sequence. Peer through a powerful telescope, and you’ll see spiral galaxies. Look at a picture of a hurricane, and you’ll find it there, too.

Something is symmetrical when it is the same on both sides, right? Well, we see plenty of examples of this out in nature! Let’s take a look at the monarch butterfly. 

Look at each of the features on the butterfly’s wings below. See how they are identical on each side? A butterfly’s wings are (usually) perfectly symmetrical. 

This is true of many of nature’s creatures: split them straight down the middle, and aside from a few lumps and bumps, they would likely be symmetrical.

Math in Nature Activity 3: Symmetry in Nature

Practice creating symmetry on your own by completing the other side of the animal in the picture.  Remember: ‘symmetrical’ means the same on both sides, down to the last spot!

Have you ever looked through a kaleidoscope and seen a ton of different shapes spinning within other shapes? Well, think of a fractal as just that: shapes within shapes within shapes.

A fractal is a repeating detail or pattern that gets smaller and smaller as it goes, creating gorgeous images that at times seem absolutely unreal.

You might see these when you look at heads of cauliflower, or when you look at the crystals on those snowflakes we mentioned earlier!

The next time you’re out on a summertime walk, take a look around you. You can find curious shapes and symmetry almost everywhere, from pine cones and maple leaves to a head of broccoli. And once you understand that the world is made of math, it might just make you appreciate your math lessons that much more!

Ready to put your lessons about math in nature all together?  Print out the scavenger hunt and take to the woods to find as many summertime shapes as you can!

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