Problem Solving Activity

Too often, we rely on worksheets to help our students learn. Students need to be active participants in their learning. They need to explore, communicate and problem solve. Here is a fun activity appropriate for second and third graders to complete during your measurement unit. Not only does it help them practice measurement skills (weighing items and counting money), but it also facilitates math process skills such as communication and problem solving.

Materials: scales, envelopes, coins, student directions and recording sheets (attached). Note – you will have to weigh the envelopes you choose to use (the ones I used weighed 3 grams).

Overview: Students work together to determine the amount of money contained in an envelope through problem solving and application of math concepts. If students can successfully determine the amount of money within the envelopes, they are given “credit” to shop for items (erasers, pens, colored pencils, etc.) in their class store.

Class Store

It is your lucky day! Your teacher said she will give you money to shop in the class store. However, there is a catch. She will not tell you how much money you will receive. Instead you need to figure it out. You will be given four different envelopes. Each envelope holds a different type of coin (quarters, dimes, nickels or pennies) which is written on the envelope. No envelopes hold a combination of coins. You must figure out how many coins are in each envelope, how much those coins are worth, and determine how much money you have altogether. Hints are given below.

Fill in the chart completely. If you determine the right amount, your group will be able to spend it in the school store.


1 envelope = 3 grams

4 quarters = 23 grams

5 dimes = 11 grams

4 nickels = 20 grams

4 pennies = 10 grams

Complete the chart:





Work Space:

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Patterning for Algebra

Algebra used to be viewed as a class for high school students. We know realize the importance of introducing algebraic thinking early in education. Identifying and studying patterns is a significant concept to help the youngest learners develop and enhance algebraic thinking. Creating, completing, identifying and describing patterns help students expand their mathematical mind. Here are some fun patterning activities for your students.

 1)    Allow students to create patterns in a variety of ways – Have students create patterns for one another. One student acts out the pattern and chooses a classmate to complete it. If the classmate can correctly continue the pattern, she makes a new one for someone else to continue. Encourage students to be creative using different instruments, movements, etc.

2)    Manipulatives – Pattern Blocks, Attribute Blocks and Color Cubes are all wonderful math manipulatives to use to practice patterning. But don’t limit yourself to these products. Use everyday items such as buttons, crayons, pencils, erasers, stickers and anything else you can imagine.

3)    Problem Solving Activities – One problem solving strategy young students often use involves determining patterns. Here are some examples that can be solved by using patterns.

A)   In a video game, the first score was worth 10 points. The second score was worth 20 points. The third score was worth 30 points. How much was the sixth score worth? Show and describe the pattern that helped you solve this problem.

B)   At the carnival, there was a prize wheel. Each student got to spin one time to see if he or she won a prize. Spinning a “1” won a prize. Spinning a “2” or a “3” did not win a prize. Spinning a “4” won a prize. Spinning a “5” or a “6” did not win a prize. Rita spun a “12.” Did she win a prize? Show and describe the pattern that helped you solve this problem.

C)   Katie’s gym teacher was trying to get them in shape. On the first day, the kids ran 1 lap. On the second day, they ran 3 laps. On the third day, they ran 5 laps. On the fourth day, they ran 7 laps. How many laps did they run on the tenth day? Show and describe the pattern.

D)   The students in the class were lining up for the music concert. The teacher lined them up 1girl, 2 boys, 1 girl, 2 boys. If the teacher continued with this pattern, would the 10th child in line be a boy or a girl? Show and describe the pattern that helped you figure out the problem.

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Pass The Times Tables Test!

Here’s a game that makes learning fun! Instead of giving your students yet another worksheet to complete, take those same problems, write them on index cards and turn it into a game. Practicing this way keeps your students focused, on-task and alert.


1)    Before class, create numbered index cards with the problems you want the students to solve (see figure 1). Create one card for each student in your class.

Figure 1 

2)    Set up the classroom. I find that it is easiest to set the desks up in a circle so that students pass the cards clockwise and there is no confusion about who to pass the card to. Before the game, I also make all the students point to the person to whom they will pass their card.

3)    Prepare the recording sheet. It is extremely helpful to provide students with a numbered table to record their answers. I find it is beneficial to have students put a star on the number problem that they are starting with so they record their answers in the correct box. (See figure 2 for an example).

Figure 2 






How to Play:

1)    Give each student a blank recording sheet. Then distribute the problems to solve. Each student is given a different index card to begin the game. Have the students put a star on the problem they will start with.

2)    Have the students solve the problem in front of them. Once they are done, the teacher says, “Pass” and all the students pass their card to the person next to them. Students then work on the problem on the card they just received, making sure they record it in the correct box.

3)    Continue this way until the students have completed each problem.

4)    Have the students provide the answer for the card they are left holding at the end of the game.


1)    This game is not a timed activity, but most students find it more exciting when they have a set amount of time to finish. I tend to supervise the class and when it appears everyone is finished, I start counting down from 5.

2)    Enforce the rule that students are not permitted to pass cards until the teacher says, “Pass.” Otherwise, students end up with more than one card on their desks and end up passing the wrong card.

3)    I have used this game with every subject and grade level ever taught. It is a great way to practice vocabulary, review for an exam, assess student understanding, etc.

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Ordering and Comparing Fractions

When students are asked to order and compare fractions, they almost always start by finding common denominators. This strategy is based on rote memorization and leads to little or no true understanding of fractions (and can be utterly frustrating!). Students cannot visualize the fractions. This article explains how to help your students compare and order fractions using reasoning skills, not math formulas.

There are three steps outlined below. Each step should be introduced separately, practiced and then combined with the steps learned previously.

Step 1 – Use benchmarks – Using benchmarks of 0, 1 and greater than 1 (improper fractions) help students get a general idea of the size of the fraction.

Example – Put the following fractions in order from least to greatest:

Encourage students to find those fractions that are equivalent to 0, 1 and greater than 1 first. Then identify if any of the fractions are exactly  Compare the numerator and denominators on the remaining fractions to determine if they are less than  or more than  Try to relate the fraction to real life examples. (“If I received 11 out of 12 on a test, did I get more than half the questions correct or fewer than half the questions correct?”)

Provide a simple table for those students who have trouble organizing their work.

Step 2 – Use Common Denominators – Many students think ordering fractions with common denominators is even easier than using benchmarks. Since each fraction will have the same number of parts to make the total, comparing is easy. Again, present fractions in real life situations that allow students to visualize them. For example, if you took a math quiz worth 25 points, who would get more of the quiz correct: the student who gets 24 questions correct (  ) or the student who gets 13 questions correct (  )?

Step 3 – Use Common Numerators – This strategy is a bit more difficult for students to grasp. The use of fraction towers, fraction circles and/or drawings helps students grasp this concept.

When the numerators are the same, you are receiving the same number of pieces of the object. However, since the denominators are different, the whole will be cut into a different amount of pieces. For example, imagine you are eating a candy bar. You receive one piece (the numerator), no matter what. If you are all by yourself, you get the whole candy bar. Now imagine one of your friends comes by. You want to share the candy bar; so you split it into 2 pieces (in half). What happens to the size of your one piece as you share with more and more friends?

For a pie version of this, Birmingham Learning Resources shows us:


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Developing Deductive Reasoning with Hula Hoop

Here is a fun game to help students in your elementary math classroom – develop their observation skills while at the same time practice their deductive reasoning. My students have named this game “Soup,” and we pretend that we are cooking up a delicious soup. Feel free to adapt it to your own students’ interests.

Materials: Attribute blocks(these are our ingredients) and a hula hoop (this is our pot in which to cook).


How to Play the Game:

1)    Have your students sit around the outside of the hula hoop so that they can all see and reach it. The teacher begins the game by creating a rule for the “soup” (e.g. square soup). Without telling the students the rule, the teacher places one attribute block into the center of the hula hoop, saying “This piece belongs in my soup today.”

2)    The first student in the circle chooses any other piece, places in in the “pot” and asks, “Does this belong in your soup today?” If the piece matches the rule, the teacher says, “Yes it does,” and the student gets another turn. If it does not, the student removes that piece from the center, and her turn is over.

3)    Students continue to take turns going around the circle. A student may guess the rule only during her turn. (e.g. “I think you are making blue soup.”) If the student is wrong, her turn is over. If she is correct, she wins the game.

4)    You can continue to play the game by creating a new rule or allowing the winning student to create a new rule for her classmates to figure out.


1)    Attribute blocks are excellent tools for this game because they contain four different attributes (color, shape, size and thickness). When I play with very young students, I choose only one attribute (e.g. red soup or triangle soup). However, when I play with older students, I use several attributes (e.g. thick yellow soup or small red triangle soup).

2)    When allowing students to create the rule and start the game, it is a good idea to have them whisper their “soup recipe” in your ear. They tend to forget their rule and provide false information at times!

3)    I find that continuing around the circle after a game is won keeps students from arguing about whose turn it is and gives everyone a chance to play. For example, if the sixth child in the circle correctly guessed the soup recipe, the next game starts with the seventh child in the circle.

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Primary Geometry All Around

Here’s a fun way to integrate math, technology and language in one project for your youngest learners. Mathematically, students learn about solid figures and their properties. Technologically, they use digital cameras and work with word processing programs to insert pictures, word process and format documents. Students practice writing original thoughts and grammatically correct sentences as they describe the objects they have discovered.

Start the project by having students take digital pictures of solid figures in their everyday lives. They can either do this at home or (if they do not have a digital camera) at school. The following chart lists the common solids primary students learn about and some everyday items children would be familiar with. Hopefully, your students will find numerous examples of each.




Rectangular Prisms


-ice cream cone

-construction cone

-cone used for sporting events

-party hat



-scoop of ice cream


-play block

-sugar cube


-tissue box

-cereal box

-stick of butter

-pack of gum

-juice box

-soup can


-dowel rod

-stove pipe

-rolling pin


Once students have taken their pictures, visit the computer lab so students can create their “Book of Geometric Solids.” Students can organize their books in a variety of ways. However, each picture should have one to three sentences to serve as a caption. Captions must identify the type of solid the object is and must tell something about that object. For example, the student inserts a picture of a soccer ball. He then writes, “A soccer ball is an example of a sphere. I play soccer every Saturday morning. It is my favorite sport.”

This activity not only allows students to discover math in their everyday lives, but it also helps develop multiple academic skills.

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Math Manipulatives in Middle School

Most teachers are very comfortable using manipulatives with their youngest students. However, fewer and fewer manipulatives are utilized as students enter middle school, and rote memorization of rules becomes the focus. Yet, Middle School Learners benefit from the use of manipulatives just as much as younger students do. Students find that decimals, fractions and integers are significantly different from the whole numbers they have worked with up to this point. Manipulatives help students explore these new concepts, communicate their thoughts, share examples and truly understand mathematics. Below are three manipulatives every middle school math teacher should be utilizing to help her/his students reach their full potential.  (Pic: Cusinaire Rods in a Staircase Arrangement;  In Wikipedia.  Retrieved March 26, 2013 from http://

Base-Ten Blocks – Yes, everyone uses base-ten blocks (units [1], longs [10] and flats [100]) with primary students to help them learn place value, counting and operations.

However, base-ten blocks are the perfect tool to teach decimal places as well. Simply reversing the value of each piece (flats [1]; rods [ ]; units [ ]) allows students to explore smaller numbers in a hands-on fashion.

Fraction Towers– Fraction towers are possibly the best manipulative for middle school students. Fractions are often the most challenged of all middle school math concepts, usually because students do not understand fractions. They are taught to memorize rules, which make no sense to them. Towers allow students to easily compare fractions and complete various operations. Working with fractions in a concrete, visual and hands-on way makes fractions less intimidating for students.

Color Chips – Color chips are most often used for statistics and data analysis topics. They are wonderful tools for those topics. However, another great way to use them is with integers. The red side is negative, while the yellow side is positive. You can even take a permanent marker and draw + and – signs on the chips. Students can model adding, subtracting, multiplying and dividing using the chips.

You don’t always need to have the physical materials. Visit the National Library of Virtual Manipulatives at: to see online versions of these and many other math manipulatives.

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Geometry Math Stars for Christmas!

Learn and Practice Geometry by Making Your own Family Christmas Tree Star Ornaments!  Great for decorating your Christmas Tree or for making a Gift to give to a friend or family member.

Materials Needed:

Colored Markers or Crayons

Glue or Scotch Tape


Colored Yarn or String


Colorful Recycled Paper (tissue boxes, flyers, old wrapping, etc.)

Directions for Christmas Star Cutout:

1.Cut out the outline.

2.Cut along all heavy lines.

3.Score plain lines on the front.

4.Score dotted lines on the back.

5.Fold triangles upword along plain lines.

6.Fold triangles downword along dotted lines.

7.Glue or tape tabs to form small tetrahedrons.

8.Continue until you have your Geometry Christmas Star Tetrahedron.

Everyone has their own Star!  Everyone has their own Inner Light!  With Favorite colors, draw your stars, or print this page and cut the pattern of the star out.

Inside each shape on the side of the star, write your name and birthday.  Or – write the names of each member of your family (If you Wish, your family / family Tree Star can be made of relatives (close or extended), friends and/or adopted family – as long as you write each name on each.  You could make many stars – a pretty star for each member of your Christmas Tree, or write everyone’s name on the same Star.

Color and Decorate each one with colored markers, crayons, sparkles, gluing pieces of recycled Christmas paper from last year.  You can also write happy words all over your star like Love, Divine Wisdom, Infinity, Pure Spirit, Fun and Harmony!

Remember to Decorate your Stars!

Glue Yarn / Strong at the top  into a 2 inch loop and tie a knot at the end

Hang your Christmas Geometry Math Stars and Decorate your Tree at Home or in your Classroom!

Have a Merry Christmas and Happy Holiday!

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Reference for Star Tetrahedron Geometry Template: “The Ancient Secret of the Flower of Life,” Vol. 2, by Drunvalo Melchizedek

Drawing with Pythagoras

“Pythagorean Theorem?!!”…I know what you are thinking…..(You can say to your class)…..”OMG  Teacher…..What could we possibly ever use this for, in the future of our lives?“……Well, in this article, we talk about real life examples of using the Pythagorean Theorem.  First let’s learn a basic calculation, corresponding to the diagram here to the <<left, followed by a bit of Pythagorean history.  The basis of the theorem is that the Area of the 2 Squares of the 2 Lines that form the Right Angle of a Right Triangle are the equivalent ( = ) Area of the Square that opposes them in the Triangle. If you look at the diagram here to the << left, the Areas of “a” added to “b” = the Area of “c” (the Square of the opposite line (or “leg”). That opposite line (or  “c” “leg”) is called the hypotenuse. Also, if one of the lines “a” of the right triangle is 4 inches and the other line “b”  is 6 inches, we can calculate how long the hypotenuse is, or the “third leg”. Letting a = 4, b=6, and c= the length of the hypotenuse.  (4)^2 + (6)^2 = c^2. Accordingly, 4 x 4 =16, and 6 x 6 = 36. Thus, 16 + 36 = 52. The square root of 52 is approximately 7.21 , hence the length of the hypotenuse or “third leg“ of the right triangle, is 7.21 inches.

The Pythagorean Theorem is named after the Greek mathematician Pythagoras. Many believe the first discovery and proving of this ancient math theorem came before Pythagoras, but since no tangible account of this has yet been documented, it is named as such. If that was true, however, we wonder what another name of the theorem would have been, and from what country and nation? (a fun question to ponder).  The Pythagorean Theorem can be used with any shape and for any formula that squares a number. And, in fact, the area of any shape can be computed from any line segment squared.  (reference also for Diagram Above).   Apparently, even the teenage Brainiac Lisa Simpson from The Simpsons television series knows all about Pythagoras.  On this site, there are some cool diagrams showing the differences of Lines (Segments / Legs), Radius, and Area:  And, to assist in figuring out these math scenarios, this site offers tabs for entering in numbers to calculate the square root of the numbers in question:

Finally, here are a few Real Life Examples:

Meet Me at The Corner:
Let’s say Stephanie and Maria are meeting at the Hiking Trail Entrance on the corner of Saanich Rd. and Cedar Rd. One phones the other on her mobile phone and asks, “How long will it be before you arrive at the entrance?”… “Well, let’s estimate by first finding out how far away we are from each other.” In present time, Stephanie is on Saanich Rd. to and is 10 miles away. Meanwhile, Maria is on Cedar Rd. and is 4 miles away.  How far away from one another are Stephanie and Maria?  The distance between them = a^2 + b^2 = c^2 or, respectively: 10^2 (10 squared) + 4^2 (4 squared); or respectively, 100 +16 = 116 miles. The square root of 116 is 10.77. Thus, Stephanie and Maria are 10.77, almost 11 miles away from one another. Hence, they figure they will be another hour on their bicycles to meet one another to go hiking.

Firefighters Needing to Know Height of a Building:
3 Firefighters receive a call to help Ann rescue her cat Tia from the Oak Tree outside her window. The tree is about 3 stories tall, and the Tia, after chasing a squirrel, is stuck on a branch at about the height of 2 stories of her house. The height to the branch may be 20 feet, and the firefighters have to put the ladder about 10 feet away from the Oak Tree in order to go around Ann‘s shed. How long of a ladder do the firefighters need in order to rescue Ann‘s cat? a^2 + b^2 = c^2 or, respectively: (20)^2 + (10)^2 = 2^2, the length of ladder required. 400 + 100 = 500. The square root of 500 is approximately 22.37. The firefighters extend their expandable ladder to be approximately 23 feet, whereas they need at least 22.37 feet to safely reach the Oak branch. Ann’s cat Tia is rescued and All are Happy!

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Teaching Math: A Breeze when Incorporating Favorite Interests

Teaching math in the Elementary classroom can be a challenge, as all elementary math teachers know, but when favorite activities of students are incorporated into the curriculum, it can be a breeze. When students have a required learning skill to attain, and they incorporate their favorite interests, then there is inspiration – and the student becomes “self-motivated”.    This is a great First Item to address with the rounds of new students returning for the fall season.  Find out Learners’ Favorites, and keep the list in a special file.

Then, as the year progresses, if a student is having a challenge learning a particular new math lesson, Teachers can refer to the student’s personal file of “Favorites Activities List”.  At this time, then, introducing the association and how that interest relates to the new lesson.

Example No. 1:  Young Matthew enjoys playing or watching the game of baseball. That is included in their list of favorite activities.  November rolls around, and the lesson of drawing shapes in geometry arises, but Matthew is not grasping the concepts.  If looking down onto a Baseball Diamond from an aerial perspective, the shape of Square is easily seen in the formation of the 4 bases on the ball field.  As well, the shape of the bases individually, is a square.

Then show in sequence what happens when the player runs to first base, second, third and fourth, demonstrating the making of a Straight Line 4 times, and in consecutive order. Within each corner, while the player stands on the base, the player looks down at both straight lines that connect, and the player can then see a perfect Right Angle of 90 degrees.  Drawing a line across from the base to the left to the base t the right demonstrates a perfect Right Triangle.

Suddenly a light is switched on in the child’s brain, and Matthew is on the way to understanding the concept of geometry.  Not only do they understand it on paper in 2-D form, but now in 3-D form, in the context of a baseball game, in a real life scenario.

Example No. 2:
  Ashley likes archery.  Archery is included in her favorite activities list.  While imaging and practicing her archery skills, she sees concentric circles – one inside the other.  When a line is drawn from her bow to the target, she demonstrates a perfect straight line.  Hence, she has a different yet equally effective association of a favorite interest to relate to the concepts in geometry –  as Mathew’s love of baseball.  Imagine now that Susie, not only is attaining the required skills the in geometry lesson, but is also having fun while doing it, and has developed self-motivation and interest in learning math.

In these examples, both sides of the Brain are exercised, (Left Right Brain Learning and Thinking) new neural connectors and dendrites grow, and you have encouraged the growth of a healthy developing young brain.

Start the school year off right, and find out what your students’ favorite activities are. Keep the lists on file, and refer to them from time to time during the school year.  You may be surprised at the effectiveness of this subtle teaching tool.

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