“Hands on Math” is an organization of ideas and teaching
strategies learned, created and developed during my 27 years of teaching.
Students learn best when they are actively involved in the lesson. When students are using Unifix Cubes and Pattern Blocks they are 100% involved in a hands-on activity, using multiple senses, and constructing understanding for themselves. Students are making their own discoveries. This is the highest form of teaching one can possibly achieve.
M & M‘s, models and manipulatives, allow students to explore mathematical concepts in the concrete world. They then can create their own knowledge and easily move to the abstract. Everyone learns better given a good model, example, and given the time and materials to investigate new ideas. It is the teacher’s task to set the stage for this to occur.
Constructivism is a philosophy of learning founded on the premise that, by
reflecting on our experiences, we construct our own understanding of the world
we live in. Each of us generates our own "rules" and "mental models," which we
use to make sense of our experiences. Learning, therefore, is simply the process
of adjusting our mental models to accommodate new experiences.
The purpose of learning is for an individual to construct his or her own meaning, not just memorize the "right" answers and regurgitate someone else's meaning.
In order for students to construct their own meaning, they must be allowed to independently discover the concepts and knowledge, and then make this understanding their own. This method of discovery is well suited for mathematics learning.
Guided discovery, an approach to instruction and learning, will help students personalize the concepts under study, creating an understanding that cannot be matched using any other method of instruction. The teacher must guide the students toward the discovery. This can be accomplished by providing appropriate materials, a conducive environment, and allotting time for students to discover.
Guided discovery greatly impacts instruction. It is the responsibility of the teacher to ‘set’ the student up to make the desired discovery. The teacher must provide all the necessary background knowledge to lead the student to the discovery. The student must realize the method(s) to be used to make the discovery. To assure this, the teacher may demonstrate what the students are expected to do. Thus, guided discovery becomes the goal of the lesson.
Plan ahead - Have the manipulatives ready for student use.
Prepare bags or containers ahead of time.
Have the manipulatives easily accessible for the students.
Allow the students time to explore the manipulative before you use them for a lesson.
Make student rules for using manipulatives.
Maintain a brisk pace when teaching a lesson.
Allow students to make predictions and estimations.
Allow a great deal of time for exploration. Use a timer.
Use correct terminology, but do not let the vocabulary get in the way of the lesson.
Always be ready to ask questions or pose challenges for those who finish quickly.
Simply, an attribute is what makes something that thing. It is a characteristic or quality that is inherent to an object. It is a way of identifying an object. Attributes of a tennis ball may include: round, green, fits in my hand, smooth, bounces, and can be pushed in by the finger. It is very important that students are always aware of the precise attribute(s) you are talking about when using a manipulative. Clearly state, and repeat, that we are looking at its color, shape, size, or the entire group. This will help students to focus on the attribute, rather than the manipulative just being a block or building piece.
Mathematics is the Study of Patterns and Relationships
Unifix Cubes are colorful, interlocking cubes which help children learn number and math concepts. Unifix Cubes represent 'units' and link one way. They are a good manipulative for counting since they fit well in little hands and are handled easily. The ten solid colors make them quite visual for demonstrations and easily allow for patterning and sorting. The single opening for linking is a useful and a fun way for children to hold the cubes.
Counting- Students may count Unifix Cubes in a variety of ways.
Teachers may also use a second item for counting that can be matched with the cubes to be counted. For example, to count to five there would be 5 Unifix Cubes on the table and 5 Q Tips. The students could count the cubes and move them as they count, 1, 2, 3, 4, and 5. Then they could count the Q Tips, placing a Q Tip in each cube as they count, counting to five again. Pennies, cereal pieces, small candies, and thin straws will also fit inside the cubes.
Patterns -Students can use Unifix Cubes to recognize, copy, and create patterns.
Number sense- As students practice and learn to count objects, the cubes can be used to compare numbers, developing number sense. For example, the student may compare five and three by counting and stacking five cubes followed by counting and stacking three cubes. Then the student can visually compare the numbers, which one has a greater value, by looking at the model he created. The cube tray may also be used for this purpose. The tray gives the student a different visualization of the numbers allowing for more opportunities for the child to construct his own meaning. Students may also line or group the cubes in a consistent way to compare the number values.
Base Ten - The Unifix Cubes easily double as base ten blocks for the exploration of ones and tens. Give each student 20 cubes, 10 of one color and 10 of another color (e.g.. 10 black cubes and 10 white cubes). Each color acts as a group of ten when connected.
Activity - The students need to sit in groups of 5 or more. 1) The
teacher has the students practice modeling numbers by saying, “14” and having
the student make 14 by using the one color group of ten and the other color,
four ones. Repeat the numbers 11 - 19 a few times and all students will quickly master the task. Then, say,
2) Next, say “27.” When the students respond. “I can’t make that!”, tell them to work with a partner. Some will do it without prompting. Now call out numbers between 11 and 40.
3) Next, say, “54.” Many will quickly share the cubes within their group to make the larger number. To amplify the concept of tens, the 10’s should always be the same color. Now, call out the numbers from 20 to 99, having the entire group work together to model the numbers. 4) Ask, “How would we make 100?” ... “Would 100 cubes be as tall as you?” Answers will be Yes and No. “O.K. Show me” Have each group build a standing tower of 100 cubes. This can be done with the cooperation of all group members.
OperationsThe cubes are a perfect manipulative to explore addition and subtraction since they can easily be put together and taken apart.
Addition Give each student 20 cubes, 10 of one color and 10 of another color. Each color will be used to represent an addend in the addition sentence.
Activity - The concept of addition is putting together numbers to form a greater number. Use each color of cubes as each addend. 1) Say “2” and have the students make two with the same color cubes (e.g.. 2 blue cubes). Say “3” and have the students make 3 with their other color cubes. Teacher models. Now describe addition as putting together numbers and ask students to show 2 + 3. Teacher model. “How many do we now have in all?” Direct the students’ attention to the two colors, numbers, now make a larger number. Repeat, repeat, repeat.
Equality - Basic Facts As the students master the facts above they can easily explore the concept of equality. After the students model 3 + 4 = 7, have them put it aside standing up. Now model 5 + 2. Place that total next to the 3 + 4 and point out that they are equal. “Is there another way of adding 2 numbers to equal 7?” Have them make 6 + 1. Show it equals 5 + 2. Have students create their own examples.
Subtraction -Activity - The concept of subtraction is taking apart numbers to form a smaller number. Give each student 20 cubes. 1) Say “5” and have the students make five with the cubes. Say “3” and have the students take three from their original five cubes. Teacher models. “How many do we now have in left?” “5 - 3 = 2.” Direct the students’ attention to notice that there are now less than we started with. Repeat, repeat, repeat.
The tunnel - Make a tunnel using tag board. Make a group, let’s start
with a train, of five cubes. Drive the train into the tunnel one cube at a time,
subtracting one each time.
Say 5-1 = 4, 4-1=3, 3-1 = 2, 2-1 = 1, and 1-1 = 0.
Fact Families -This activity puts together the adding and subtracting described above. Have the students make a basic fact using two addends of two different colors. For example, 3 red plus 5 blue equal 8. Have them move the three red cubes from one end to the other end to show 5 blue plus 3 red equal 8. Now start with the 3 red and 5 blue equaling 8. Remove the 5 blue and say, “8 take away (minus) 5 equals 3 (red).” Now put the eight back together and then take away the 3 red and say. “8 minus 3 equals 5 (blue).” Repeat and model, then have students do a few together. Once they get it, allow them to make up their own and explain their family of facts to a partner.
Tens and Ones Regrouping- Using the Cubes linked in groups of 10 of one color, students may easily take apart a group of 10 to make ones and subtract. For example, 32 - 14 = should be modeled 3 groups of 10 and 2 ones minus 1 group of 10 and 4 ones. Students should see they cannot subtract 4 ones from 2 ones. So they need to regroup a 10, taking them apart, making 12 ones. Now they can subtract 4 ones from 12 ones. Then subtract the tens.
Multiplication -Multiplication is repeated addition. This can be easily shown using the Unifix Cubes. Many teachers lead students to the multiplication facts by practicing skip counting; counting by 2’s, 5’s, and 10’s.
Activity - Give each student 30 cubes, colors are not important. The students are going to make stacks of cubes using the number they are counting by. For example, if they are counting by 2’s, the students will make stacks of two. Then the student will count by 2’s picking up and putting together the stacks as they count. They may try to stack them in a large vertical tower, place them on their desks in an organized way, or use the Unifix grid. Using the grid allows the students to see easily the repeated numbers and the patterns the numbers will make.
Activity - The students may practice their math facts, one to five using
their fingers and the Unifix cubes. Let’s multiply by three, Have each student
put a cube on the middle three fingers of one hand. This represents three groups
of 1 or 3 X 1 = 3. Now have the students put a second cube on each finger,
representing 3 groups of two or 3 X 2 = 6. To show 3 X 5, have the students put
5 cubes on each of their three fingers. And so on.
Students may then work with a partner to continue their facts up to ten.
Perfect squares - You may use the cubes to easily show the meaning of a perfect square. Students may use the grid to show how 3 rows of 3, 3 X 3 = 9, create a perfect square.
Graphing- Use the cubes for beginning graphing activities. Each cube can represent one response, linking the cubes in a stack for the total number of each response. A different color cube can be used to show the different items being graphed.
Fractions -The cubes are well suited for developing the concept of a fraction meaning a part of a group and the concept of a numerator and denominator.
Activity - Give each student 20 cubes, 10 each of 2 colors. Introduce to
students that fractions can mean a part of a group. Use four cubes, three of one
color (blue) and one (red) of another color. You can place them, one each on top
of your four fingers of one hand. Show and explain that a fraction has a top
(numerator) number and bottom (denominator) number. The bottom number is the
total number of parts in the group. In this case, four. The top number is the
number of parts we are talking about, in this case one red cube. Write 1/4 and
make the connection between the cubes and the written fraction. Now exchange a
blue cube for another red. Now what is the new fraction for what part of the
group is red? 2/4.
Always identify the denominator first. Repeat, now making it 3/4 red.
Now the students should work with you following your directions to create and describe fractions as part of a group. They enjoy putting the cubes on their fingertips. For example, “Let’s make a group of three. How many cubes do we need? Use two of one color and one of your other color. What part of the group is the one cube? 1/3 What part of the group is the other two cubes? 2/3
If you wish, introduce 3/3 equal one whole. Now let’s make them all the same color. What fraction is this? 3/3. Repeat, repeat, repeat. Once they understand, you can just say, “Show me 2/5. What fraction of the group is the other color, the three. 3/5.
Equivalent Fractions- You can easily introduce equivalent fractions continuing the same method. Have the students make 1/3, using their two colors, on three fingers. For example, one red and two blues. Now, have them put a second cube on each finger matching the first colors: a second red, a second blue, and a second blue. They now have a total of six cubes, two red cubes on one finger and four blue cubes on two fingers. Now, “How many total parts are in the group? 6. How many are red? 2/6. Is that the same as 1/3? Yes! Yes 1/3 is equivalent to 2/6. Let’s look at the blue ones. What fraction are they now? 4/6. Is that the same as 2/3? ... Now continue the pattern adding another red and 2 more blues. Now 1/3 =3/9 and 2/3 = 6/9. Continue and repeat using other fractions.
You can also reiterate the equivalency of a whole. For example use two same color cubes on two fingers. What fraction does the cubes represent? 2/2. Two halves equal one whole. Now add one to each finger, same colors. Now 4/4 = 2/ 2 which equal one whole. An so on.
Measurement- One of the concepts all students must grasp and understand is that we measure in units. A Unifix cube is a good unit for measuring, especially for little hands.
Activity - Linear Measurement Have the students work with a
partner. Give each pair of students 30 cubes of different colors. It is
important to use different colors because the students must realize the
attribute of the cube that is now important is its length, not its color. Have
the students measure items in the classroom, and record their measurements. To
ease classroom movement during this activity some of the items should be common
to all , such as their finger, a marker, their shoe, or their math book. All
other items must be easily accessible in the classroom, such as a toy, a map or
poster, a board eraser, or a file folder. To start, the teacher introduces the
activity, being sure to emphasize the term unit. She then models how to
measure using the cubes. Finally, how to record their measurements on the
worksheet emphasizing that each measurement must be labeled with its unit, 4
cubes. When the item measured is not exactly a number of cubes tell students to
write the number of cubes it is closest to. For example a marker is between 6
and 7 cubes long. I would write 6 cubes because it is closer to 6 cubes.
For older students, they may include a fraction. The marker is 5 1/2 cubes long.
Follow up - After the students have had many measuring experiences, make a similar worksheet adding a space (column) for the students to predict their measurements.
Extensions - Area - Square units The cubes and the tray may be used to measure in square units. Students can build models using pipe cleaners, straws, or toothpicks to model their measurements.
Volume - Cubic units The cubes and the tray may be used to create solids to explore volume. Students should calculate (count) the cubic units in these models.
Extensions - You can explain to the students that the results of any one trial may not reflect the real mathematical probability because they only did the experiment 10 times. You may add together, on the board, the outcomes of all the experiments for a trial and the results will more closely reflect the mathematical probability.
Pattern Blocks are one centimeter thick multicolored blocks that come in six shapes; hexagons, squares, trapezoids, triangles, parallelograms, and rhombi. For ease in identification, each shape is made of only one color. For example, all hexagons are yellow and no other shape is yellow. The blocks may be used to demonstrate, discover and explore many mathematical concepts.
Patterns -Students must gain proficiency in recognizing patterns since it is essential in the study of mathematics. Pattern blocks allows the beginner to copy, continue, and create patterns. Since each pattern block has two distinct attributes, shape and color, they are perfect for pattern exploration.
Activity - Students must be given many blocks to work on patterns. For
beginners, or students who are having difficulty, you may limit the number of
different blocks used in the pattern and the number of different blocks
available to the student to continue the pattern. The students may work
individually or in pairs to solve patterns. The pattern may be on a card for the
student to copy. After they copy the pattern on their desk, they may continue
the pattern at least two more times. As students become more comfortable, the
pattern may be given a minimum of two times then the student may continue the
Next, the students may create their own patterns. Finally, they can create patterns for their partner to complete.
Extension - The teacher may create a more complex pattern on the overhead
and have the students look at it and then copy it. Students should not be
looking at it and building at the same time. You can show the design for five
seconds, instruct the students to build, give them time to build, then show
again for five more seconds for students to check, allow further building, then
show one more time to check.
Challenge - Create patterns that have ‘open’ areas within the figure. You can do this by building a design and then remove a piece that is surrounded by other pieces.
Geometry -The shapes themselves may be used to identify shapes and call attention to each shape’s attributes. Next, the students may explore building more shapes by putting different blocks together. The teacher may create challenges by requesting a shape be constructed of either specific blocks, “Can you make a hexagon using only trapezoids and triangles?”, or requesting a shape constructed by a specific number of blocks, “Can you make a trapezoid using any three blocks?”
Symmetry - Symmetry is an attribute of a shape or relation; where there is an exact correspondence of form on opposite sides of a dividing line or plane. Examples of symmetry are found in many animals (vertebrates), designs and patterns, and the letters of our alphabet.
Activity - Students must be given many blocks to work on symmetry. The
students may work individually or in pairs. First the teacher must demonstrate
using the overhead and pattern blocks. The teacher may start with the hexagon
and add one block to its side, e.g. the trapezoid. Have the students copy the two blocks on their desk and
ask them what needs to be added to the other side so each side looks the same. Keep it simple, adding only one block at a time. Once students have mastered
this, the teacher can demonstrate a bit more complex design, only creating one
half of it. Then have the students copy what is there and create a symmetrical
Students may then use this method to create large beautiful designs.
Multiplication -The blocks may be used to introduce and practice the concepts of multiplication.
Equations -Students may begin to work on equations continuing the above methods.
Fractions -Use pattern blocks to introduce fractions to students. Once the teacher explains the fractions using the overhead blocks, students can explore a whole, halves, thirds, and sixths. When talking about fractions, a part of a whole, it is important to identify clearly what is the whole. To begin, use the hexagon as one whole.
Activity - Students must have at least one hexagon, three rhombi, and six
triangles. More blocks should be available. The teacher first writes the word
“whole” and discusses with the class what a whole means to them. Then the idea
of fractions is introduced, being a part of a whole. On the overhead the teacher
then displays the hexagon and states, “This is the whole.” Students follow at
their desks. The teacher then asks, “Can you make more wholes, yellow hexagons,
using only one color to make it again. Go ahead and try.” Students work to move
like pieces to duplicate the hexagon. Students having difficulty may be prompted
to build the new whole on top of the hexagon in a puzzle-like fashion. Allow
students plenty of time to explore. Those who finish quickly can be challenged
by asking, “Are there other ways the pieces can fit into each other. always
using just one color.” After all students have built more hexagons the teacher
then discusses and names the fractions. As this is done, the concepts of a
numerator and denominator are revealed, though the terms themselves need not be
mentioned. “Let’s look at how we made more hexagons. How many red ones make the
whole? [Introduce how fractions are written.] Two. Each one is called one half.
[write 1/2] The bottom number is the total number of pieces needed to make the
whole and the top number is how many we are talking about, one over two, one
half. To make the whole we must have two halves, 2 over 2, 2/2.” Repeat this for
thirds and sixths. At this point the concept of fractions of a whole may be
emphasized. 1 whole = 2/2 = 3/3 = 6/6. “Is 6/6 the same as 2/2?” And so on. The
teacher can then compare the concept to puzzles, “If we have a 20 piece puzzle,
how many pieces do we put together to put it all together, to make one whole?
... 20/20 = one whole. Have the students find other examples. [crayons in the
box, chapters in a book, days in a school week, or egg cartons]
Extension - Fold paper to show fourths and halves. The paper is one whole. Fold the paper in half and emphasize two halves make a whole. Then fold it in half again and show four fourths, four fourths make a whole.
Next - Repeat the above demonstration, using the trapezoid and rhombus as the whole. Now we can start naming parts of the whole using various fractions. “Let’s look at the thirds, the 3 blue diamonds, rhombi. Show me one third, explaining the concepts of the numerator and denominator once again. Compare the values of each fraction. “Which is more, 1/3 or 2/3? Which is more 2/3 or the whole. 3/3?” Continue using the thirds and sixths.
Comparison and Equivalent Fractions - Now we may compare each of the
names for a whole with each other. Make the models of the whole, one hexagon. Review the names and meanings of each fraction as a part of a whole. Now
let’s look at each whole and compare them to each other. Which is bigger 1/2 or
1/3? 1/3 or 1/6? This shows an important concept of fractions, as the
denominator gets larger the part of the whole is smaller. [Compare it to jigsaw
puzzles.] Next compare other combinations. Which is bigger 2/3 or 5/6? 1/2 or
2/3? And so on.
Finally, start comparing equivalent fractions. Which is bigger 1/2 or 3/6? 2/3 or 4/6? And so on. “Can you find some more equivalent fractions?” Remind them that 1/1, 2/2, 3/3, and 6/6 are also equivalent fractions,
RatiosUse Pattern Blocks to introduce ratios. Compare the block to its number of sides, For example one square has four sides, 1 to 4 ; the hexagon, 1 to 6; and the triangle, 1 to 3. From there you can set up proportions. one hexagon to six sides, two hexagons to 12 sides, three hexagons to 18 sides, and so on.
Game - Last Turn WinsThe object of this two player game is to be the player to place the last pattern block into the shape. Printable Note Due to printer variances, you may have to enlarge this printout by 5% - 10%.
Interactive Pattern Maker http://standards.nctm.org/document/eexamples/chap4/4.1/index.htm#applet
Pattern Blocks NCTM http://illuminations.nctm.org/mathlets/patchtool/index.html
Pattern Blocks http://ejad.best.vwh.net/java/patterns/patterns_j.shtml
Adding and Perimeter http://www.edb.utexas.edu/faculty/wstroup/Gen_Act_Web/I03/zga03.html
Creating, Describing, and Analyzing Patterns http://standards.nctm.org/document/eexamples/chap4/4.1/Part2.htm
Counting Groups Preliminary Division http://www.edb.utexas.edu/faculty/wstroup/Gen_Act_Web/I14/zga14.html
"How Do You Measure Up?" - Shadow Study http://www.naturenet.com/workshop%20homework/newpage3.htm
Measuring Length with Printout http://cesme.utm.edu/resources/math/MAG/K-2MAGActivities.pdf/K-2C2A3.pdf
Problem Solving Activity Mr. McGregor’s Garden
There’s More Than One Way To Answer That Question Addition Combinations
Block Designs with Printouts http://www.eduplace.com/math/mathcentral/gradeK/koca2.html
Create Pictures http://www.edu.pe.ca/westwood/MathArtWeb/pattern_block_pictures1.htm
Fraction Shapes http://math.rice.edu/~lanius/Patterns/notes.html
Investigation Fill in A Region http://www.terc.edu/investigations/curr/HTML/men1.html
It’s A Perfect Fit Identify Numerical Relationships
Pattern Block Activity Shapes are given values http://www.st-james.richmond.sch.uk/year4math.htm
Pattern Blocks MANY ideas and activities http://fcit.usf.edu/math/resource/manips/pattern.pdf
Last Turn Wins Printable Note Due to printer variances, you may have to enlarge this printout
by 5% - 10%.
Unifix Cubes Probability Activity Printable