Children often learn math best with concrete, real-life, physical examples that they can feel, touch, move and manipulate. Educational suppliers have long known this, and sell many different kinds of math games and manipulatives. Many of these you can make easily and inexpensively yourself.
Math in Everyday Life
Laundry: Yes, laundry can be a math activity. Before washing, practice sorting clothing by colour, fabric etc. After washing, sort laundry by owner and type. How many socks are there? Practice counting by 2's. Who has the most blue items? Who has the least?
Cooking and Baking: These are great for working on measurement (volume, temperature and time), ratios (try doubling or halving a recipe, etc.), division (serving portions) and also doing a little kitchen chemistry: What ingredients make things rise? What adds moisture? Allow for some experimentation when possible. Check out the edible science page and kid-friendly recipes for more inspiration.
Running Errands: At the grocery store, compare prices based on volume and weight per unit; compare nutritional labels and serving sizes; calculate the cost of ingredients for a specific recipe; estimate the cost of the grocery bill before reaching the checkout; compare distances different foods have traveled to reach your table. For any shopping trip: try using cash for your transactions and share how to make change with your child as you pay for items. Practice calculating sales tax together. Compare package size and actual volume/weight of various items since sometimes what appears to be "more" is, in fact, not. Plan the most efficient route to take for trips with more than one stop. Use online flyers to compare prices to determine which store will meet your shopping needs most economically based on your shopping needs.
Craft sticks (popsicle sticks): Use these to make homemade cuisinaire rods by painting and cutting to appropriate lengths; also use them to show regrouping by bundling up groups of 10 and fastening together with an elastic band (for base 10--you can do this with other bases too), make "superbundles" of 100 by bundling 10 bundles together in a similar way. Subtract by undoing bundles as needed.
Jack's Beans: Use large dried beans for this project. Spray paint one side only of your beans. Now you are ready to explore probability and number sentences. Choose a number to work with, say 5. Toss out 5 beans and count those with white and those with coloured sides. Make a number sentence with them, say, 4 + 1 = 5. Or, simply count the coloured sides and throw multiple times. Record the results of each throw, then plot the results on a graph.
Candy Math (or Pasta Math): For this activity you will need either candy or pasta with different attibutes, such as colour, shape, type, flavour, etc. Fruit gummies work well because you can get them in various shapes and themes, ie. dinosaur shapes mixed with animal shapes can be sorted by extinct and living, by individual species and by colour. Halloween candy also lends itself well to this activity. If you choose to use pasta, look for a variety of shapes, sizes and colours in your selection. Pasta can be grouped by size, colour, shape and whether it can be threaded onto a string or not. If you don't have a great variety on hand, try buying in bulk so you can pick and choose how much and what types you would like to use.
You may wish to start with addition, adding and counting out the items. Division can come next (sharing the candy evenly between friends, stuffed animals, etc.) and multiplication by reversing the activity. You can sort by the various attributes, make Venn diagrams using needlework hoops, hula hoops, or lengths of string tied into circles. How many of the red ones are also dinosaurs? How many of the shells are orange? Try graphing by attributes as well--you can draw a graph or us the actual materials to form a graph on your work surface. Subtraction comes last, and if you are using candy you will subtract by eating them of course!
Playing Cards: These are great for matching, addition and subtraction games and studying probability. You can use games you know, search online for new games, or try one of these:
- Addition game: Pick a number, say, 13, before starting the game. Each player is dealt 7 cards. Players may put down sets of cards that add up to the chosen number. Face cards count as 10, aces are high or low, etc. The goal is to be the person with the most sets of the given number. To take turns, players may either select a card from the deck or ask another player if they have a certain card. If the player asked has the card, they must hand it over.
- Multiplication and division game: Deal each player 5 cards. Turn over the top card: this card will be the X card. Player 1 puts down a card. The card value is multiplied by the face card and the total is recorded. If the X card is a 3, and player 1 plays an 8 this will give a score of 24. If a player is caught making a multiplication or addition error, they must divide their score by the last card they played. Player 2 takes a turn in a similar way, until one of the players reaches a score of 500.
- Variation: assign one colour of cards to be a negative integer, ie. in our example, if red cards are negative, and a 2 of hearts is played, the score for that turn will be -6.
Manipulatives to Create Yourself
Base Ten Set: Use construction paper, tag board, bristol board etc. to make a base 10 set. For single units, cut out 1 cm square pieces. For 10's, cut out strips 1 cm x 10 cm. For 100's, cut out a 10 cm x 10 cm square. For a 1000 cube, make a cube that is 10 cm x 10 cm x 10 cm. You can use the template in the picture here, or make your own, although it's even better if the learner builds the set to practice measuring.
To make a cube, trace a column of four 100 squares. Add another square on the top left side and another on the bottom right side. Add a 1 cm gluing flap along the outside edges. Cut out and carefully fold along the lines you have drawn. Glue together using the flaps.
Base 10 sets are great for measurement, base 10 activities, and showing exponential growth.
Pentominoes: These are arrangements of 5 square units joined along the sides to make various shapes. These shapes include: 5 in a row, which looks like a lower case l, 4 in a row with one at the side at the bottom, 4 in a row with one at the side one up from the bottom (and also flips of both of those), 3 in a row with one to the side at the top and one at the other side to the bottom (an "s" or a backward "s"), a "u" which is an "s" with both on one side, a "T" and a "t", and a double row of two plus one to the side. See how many ways you can make a square of various sizes with the shapes. Try making different pictures. Invent challenges using a fixed size, shape and/or number of pieces. This is similar to the game "blockus".
To make your own set, trace out the pieces (1 cm squares work well) and cut them out from construction paper, bristol board etc. Keep the pieces together in an envelope or sealable freezer bag when not in use.
Tangrams are a series of shapes that are cut from a square in a specific way. The first challenge is to try and reassemble the shapes into a square. Can you make a rectangle? How many different animal shapes can you make?
You can buy tangrams commercially or copy the pattern on
paper and make your own.
To fold your own, start with a square. Fold in half, corner to corner and cut or tear along the folded edge.
Follow the folding images above, folding and cutting or tearing along the dotted line in each.
You will end up with two large triangles, one medium triangle and two small triangles; a square and a parallelogram.
You can also print the blackline image at the top and simply cut out the shape.
The following site has some interesting history and challenges regarding the tangram. You will need to scroll down a bit to get to the good stuff! Click here for the Absolute Astronomy Tangram Page
Also, here (on page 6) you can find the legend of the tangram.
Fractions can seem confusing to some students, but really all they are is mini division questions. You can also view them as little puzzle pieces that when assembled, come together to create a whole.
You can start introducting fractions by using everyday objects around the house or classroom. What makes a good example of an everyday fraction set? These are my criteria:
- It must have equal parts that form a whole that can be somehow identified (floor tiles of a whole room, or ones in the area marked off with masking tape, for example).
- It must be present, measurable and real in order to be considered concrete (time is not so concrete, but a calendar or analogue clock face can be).
Some examples will be easier to use than others. If you choose to use an egg carton, use marbles, pebbles, beans or other place holders to show what parts are used.
Using food containers alone you could probably find examples of halves, thirds, quarters, fifths, sixths, eighths (hotdog buns), tenths, twelfths (hotdogs), 20ths, 24ths, etc. Some quantities seem to be more common than others, which is another cause for classroom discussion.
There will likely become a time not far into the discussion of fractions where some smaller and more manageable examples will become more practical. Once you have exhausted examples of fractions in everyday life (ice cube trays, orange sections, pizza etc.), you may find you want a set of manipulatives you can work with on a smaller and more convenient scale. Here I've made a kit of rectangular fractions from paper. To make this, start with a rectangle shape. To reduce the amount of work involved, have students cut four of these in one go. Give each student four sheets of different coloured paper (using a sheet of 1 cm graph paper as one of these sheets can help provide a stronger visual representation). Show them how to use paper clips to hold the pieces together in place as they cut. Once they have measured and cut their pieces, they should have four rectangles of the same size but of different colours.
Now choose one colour (usually centimeter graph paper, but you can vary this if desired) to keep whole as a base, and the other three to divide into equal parts. After dividing each colour, stop and label those pieces before moving on. For example, if you divided the yellow sheet into halves, label each half as 1/2 before moving to the next sheet in order to avoid confusion. Using a paper clip to hold all of the same sized pieces together also helps keep things organized. Store each set in a large envelope to keep all the pieces together.
Of course, you may decide you want more than those fractions, so simply repeat the process by tracing your base rectangle and cut out more fractions as desired. You can also make circular, square or other shaped sets in a similar manner.
For more ideas regarding working with fractions, see my blog post on the subject here.
Pyramids, Squares & Cubes
small square tiles (paper, ceramic, wood etc.) of uniform size
small cubes of uniform size (dice, sugar cubes, wooden or plastic cubes, etc.)
Purpose: to explore and build number patterns with tiles and cubes.
As with all other manipulatives, it is best to encourage students to explore with the materials for a while and find their own patterns with them before providing them with additional challenges. I've detailed a few challenges below, but when working with students on these, it is better to ask them more general questions and let them find their own patterns and solutions.
Pyramids: use your tiles to build 2-dimensional "pyramids". The top row will have one tile, the second row will have two, the third will have three and so on.
Write down the sequence of both each row, as well as the sum of all the tiles up to the end of each row. What can you determine from this pattern? Can you use this to predict the sum of the 10th row?
For most students, you might just show them the first two or three rows then ask them to predict the 10th or 12th, challenging them to try it without using the tiles if they can.
Now try a slightly different pyramid. For this one, your top row will have one tile again, but your next row will have three tiles. Each new row adds a tile on each side as you move down. Record the rows as you did above. What patterns can you see emerging? How can you use this to predict the numbers for subsequent rows?
Students who enjoy this activity might also enjoy working with Pascal's Triangle.
Squares: Start with one tile. Notice how it is exactly one unit wide and one unit long. Add tiles until you have a square that is 2 units long and 2 units wide. How many did you have to add? How many are there in total?
You can write this as 2 x 2 or also as 22 for which we say "two squared".
Try adding tiles to make 32, 42, etc. What do you notice about the total tiles needed for each? What pattern do you follow when adding the new tiles?
Cubes: You can follow a similar pattern to make cubes. Start with one cube. Notice that you can measure it in 3 directions, height (or length), width and depth. Each of these measures one unit. This can be written as 1 x 1 x 1 or 13. (If you have already discussed squares, ask them to guess how you might write this as an exponent rather than just tell them). The 3 here means that you multiply the base number by itself three times. The 3 is also called an exponent.
Try adding more cubes so that you have a cube that is 2 x 2 x 2 or 23. How many cubes did you need to add? How does this compare with how many tiles you needed when you used tiles to make squares? Continue building cubes with higher numbers, and look for more patterns as you build.
To make these, either print the template on paper that has different colours on each side, or print out a copy then glue a contrasting sheet of coloured paper to the back. Cut out the pieces along the black lines. One colour will represent positive numbers and one will represent negative numbers. You can use the template above, or print from this PDF file (now with thinner lines).
Algebra tiles can be used in a variety of ways to show concrete representations of equations and expressions. For example, for the expression
2x2 + 4x + 6
you can use two of the large squares to represent 2x2; 4 strips to represent 4x, and 6 small "unit" squares to represent 6. From here you can factor the expression or otherwise manipulate it in a concrete/visual manner.
For the expression
x2 - 3
use one large square for x2; and 3 of the unit squares with the negative side up for the -3.
Homemade Strategy Games
Mancala Game: For this you will need a clean egg carton (use a plastic or styrofoam one that can be thoroughly washed out) or an ice cube tray that holds 12 ice cubes. This will form your playing board. You will also need an assortment of marbles or pebbles as game pieces, and a small box or container for either end of the carton/tray that will serve as your pits.
To play: 3 stones (or whatever you are using) are placed in each section of your board but the pits are kept empty. One player picks up all of the stones in any one of the holes on his side. Moving counter-clockwise, the player deposits one of the stones in each hole until the stones run out.
If the player reaches his or her own pit, her or she deposits one piece in it. The opponent's pit is skipped.
If the last stone a player drops is in their own pit, they get a free turn.
If the last stone a player drops is in an empty hole on their side, they capture that stone and any stones in the hole directly opposite.
Moves after the first one continue in the same way, with the player removing the stones from any hole and distributing one to each hole in a counter-clockwise direction.
The game ends when all six spaces on one side of the Mancala board are empty.
The player who still has stones on his or her side of the board when the game ends captures all of those stones.
The player with the most stones in his or her pit is the winner of the game.
Another bean game that is often mistaken for Mancala is Bao. This game comes from Zambia and has a strategy all its own. The board is a 4 x 8 series of sections (similar to Mancala) and beans are distributed around each player's own side, with rules for capturing beans from your opponent based on the comparative number of beans in adjacent sections. The goal is to capture your opponents beans until they cannot make a further move in the game. More detailed description with full rules can be found at this link
The strategy involved in this graph paper and pencil game is similar to that used in the popular game "Blockus".
This game uses graph paper. Each player uses a different colour or symbol. Starting at opposite corners of the page, each player colours in (or writes their symbol inside) two boxes on their turn. Those boxes must share a side when they form a new "block", and after the first turn, they must touch another set of that player's boxes by touching only at a corner (the new block must not share a side with an older one of the same colour or symbol, which is similar to the game Blockus). The players alternate turns, and the goal is to be the last one able to place a new box.
Boxes of different players may share a side as they pass each other, but must not overlap boxes.
Variation 1: Each player starts with a single box, and on each successive turn adds an extra box to their block. This means that on the 3rd turn, the block they add will have 3 boxes. Those boxes must each share a side with another box, and could be in a row or an L-shape, as long as they fulfill the one-side connection rule. As the blocks grow, so with the potential for different shape configurations. This game moves much faster than the first version.
Variation 2: Using either of the versions above, the goal is now to reach a square touching the opponent's original block before their opponent reaches theirs.
Click here for a free printable file-folder version of this game.
Dots Game: All you need to play this 2-player game is a sheet of paper and pencil. Decide on the size of your grid and draw dots in vertical rows that also line up horizontally (if you use graph paper for this, the dots would be at the intersecting corners). The first player connects two adjacent dots with either a vertical or horizontal line. The second player connects two other dots in a similar way. The players take turns doing this until a box is formed. The player who closes a square (by adding the line that completes the 4th side) puts their initial in the box and gains a point. The play continues until all of the dots have been joined to make boxes. The winner is the person with the most initialled boxes.
Alternative: Decide ahead that the winner will be the person with the least completed boxes.
Variation: Offset alternate rows of dots and play "triangles" instead--same rules, except instead of forming boxes, you form triangles to score points.
More math games to make can be found at these links on the next page:
Battleship Coordinates Game
Code Breaker (like Mastermind)
Black (road making game)
Bird Cage (a different path making game)
Labyrinth Pencil Game
Hexaflexagons are interesting paper crafts that combine geometry with origami. To make one, you need to start with a narrow strip of paper.
Fold it into a series of equilateral triangles. We tried this several ways but found that twisting it, as in the picture to the left, was the most effective way to do this and still account for the thickness of the paper.
Now you need to keep folding these triangles along your strip until you have 9 triangles (10 if you wish to use an extra flap to glue it together).
The next part of the operation is folding the strip into a hexagon. Take the first three triangles and then fold the rest of the strip over top. Repeat with the second three triangles. The exposed edges should look something like the diagram on the right. Be sure that as you move your finger around the hexagon shape, each time you reach a place where the paper overlaps, it forms a pocket. It doesn't matter if this happend for a clockwise or counterclockwise circle, as long as it is consistent.
If you are using 9 triangles, tape the edges of the end triangles together; if you are gluing, fold over the remaining triangle and glue it. Fold your hexaflexagon both ways along each fold that connects opposite corners so that it will move smoothly from one position to the next. Now, pinch the alternating folds and open it over to the next side. Repeat. Repeat. (Warning: this can become highly addictive!). Colour it with markers, pencil crayons, crayons, stickers, etc. as desired.
The topology of hexaflexagons is that of a mobius strip. To find more activities relating to topology on this site, click here.
Here are some excellent videos and links related to hexaflexagons, triflexahexagons and other flexagons:
Vi Hart Videos on Hexaflexagons:
Hexaflexagons Part 1,
Hexaflexagons Part 2,
Hexaflexagon Safety Guide
A Bestiary of Flexagons
Flexagon.net Complex Flexagons with diagrams
Martin Gardner on Flexagons
Pumpkins lend themselves readily to many mathematical investigations. The seeds, when cleaned, make excellent counters, or you can colour one side and use them for statistical games as mentioned above. Here are some more ideas to add to your mathematical bag of tricks:
- Predictions: Will it float? How much does it weigh? Which pumpkin is heaviest/lightest/fatter/? Which pumpkin has the most seeds? How many seeds with the pumpkin have? How many days will it take to decay?
- Comparisons: graph the numer of seeds from several pumpkins and compare, compare weights and circumferences of pumpkins, compare the thickness of the skins.
- Use measurements of the pumpkin to try and calculate Pi and/or Tau. If you use Pi, you just might have to make some pumpkin pie!
- Make pumpkin pie together by measuring ingredients, temperature, baking time, etc.
- How far can you roll the pumpkin on a single push? Which pumpkin will roll the furthest?
- Plant several seeds and compare their growth. Maintain a graph to show the progress.
- Compare the prices of pumpkins between various farms and stores. When is it better to buy by weight vs. a set price per pumpkin? Express this using algebra and geometry.