Handful of Blocks

 

This activity is for a pre-kindergarten to 1st grade level student to organize, interpret and translate unifix cubes to a three dimensional  bar graph.  It is probably the first time to make this construction and perceive this type of organization in a meaningful way.  

Objective:

1.   Students will collect unifix cubes of four different colors by a "handful" from their bag and sort to container.             
2.  They will match their cubes colors to a construction paper of the same color making a tower of the same
 color from their "handful". 
3.  This will continue until all four colors are placed with the matching color cards.                
4.  They are connected and comparisons of like and different towers are built are constructed.
5.  Students will finally be able to interpret their learning in a two and three dimensional graph transferred to
 paper.

Here are three sheets out of four without the color word ( examples of color cards) described in this post.

Prior Knowledge:

                 1.  Students should be able to count to ten objects
                 2.  Students can identify the numeral 0 through 10
                 3.  Recognize colors
                 4.  Make comparisons between taller, shorter, more than, less than, fewer.

Materials:  Each student  ( Preferable 4 in a group)

                1.   Twenty unifix cubes of four different colors (five of each) in a bucket or large plastic bag
                2.   Four different construction sheets that match the four unifix cubes (will have color words
                      written on each matching sheet)
                3.  crayons
                4.  "Handful of Blocks" Vertical Graph
                5.  "Handful of Blocks" Horizontal Graph
                6.  16 small containers ( 4 for each color, for each child to sort cubes in the beginning)

Experience

Have the four children encircle you.  Tell them them that today we are going to play an activity called,  "Handful of blocks"    Hold the materials while displaying a group of four cubes to the group, one group of each color and call on a students to identify the colors of the cubes.

Instruct the students spread out and show them the color cards and the names of the colors on the card.  Have them lay out the cards.
After that task, each child will receive their bag of 20 unifix cubes of 4 colors, 5 each. Guide them to "Grab a Handful of Blocks!"  Students will probably have some spilling out but encourage them to keep out those that were in their hand.  Have students sort the cubes by color into each of the four containers.

Investigate

Have one of the students take a handful of (e.g. red) cubes while the other students pick other colors.  Find the color card or paper that matches the color of the cubes.  Have the students count the number of their cubes.  Place the color cards side by side and have the students continue to connect the cubes of matching color.

 Have the students place the cubes vertically or stand them up in front of the matching color card.  Explain to the students that they have created a "bar graph".  Those bars or trains of cubes you made make it easier for us to compare the number of cubes in each color.       

This is similar to what the students could be seeing in the bar graph.  Only we have described using 4 (four) colors.

 Questioning

1.  How many orange cubes do you see? 
2.  Are there more red cubes than orange cubes?  How did you decide?  Did you count?  Can you decide
     without counting?  How?  Yes, you can compare the cubes.
3.  The green bar is taller than the red bar.  What does that mean?  Yes, the green bar has more cubes than
     the red bar.
4.  Which bar has the most cubes?  Yes the blue.  Did you need to count?  Which bar has the fewest           cubes?  Yes, the yellow.  Did you need to count?  No, you could tell by looking at their size. 

Change the vertical orientation to horizontal keeping the color cards oriented with the corresponding cubes. Turn the graph so that the bars are not taller but now are longer  bars.  Do the same answers apply?  Continue the investigation with the students.  If possible, tape the horizontal graph up for further reflection.

Analysis

 "Handful of Blocks" is an exercise that enables teachers to view where students are in counting and
making collections.  This is also a meaningful construction of a three dimensional graph to represent their  learning in a small group setting.  The math language development of more, less, fewer comparisons becomes apparent.

Comparing the vertical and horizontal graphs is a prelude to measurement.  Always set the trains or towers the same so that there is a common threshold or starting point for comparison.
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Square Boxes

From, Coming to Know Number, G. Wheatley, 1999

 The objective of "Square Boxes" or math squares is to enable students to make sense of adding and subtracting for the young learners.  These squares provide explicit ways for learners to construct their own way to make these operations more concise.

As in the scale activity, we have no operational or equal sign.  Students have to examine the boxes and base their actions on logic.  Instead of rote algorithms, the students must search the numbers and determine what to do with them.  There are always multiple ways of progressing.  The numbers are chosen to be compatible using ten as the standard base.

Introduce math squares as a whole class activity.  In the above example the teacher could direct the students by saying, "We have four rooms in this box.  We can see how many people are in each room.  How many people are there altogether?

Students are inspired to talk interactively.  Observe to see it they are using mental imagery of their tens frames.  Instigate students to discuss their different strategies to find the sum.

Another type of square may have a total at the bottom, but one or more rooms left empty.  The teacher may say, "Here we have four rooms.  We know how many persons there are altogether.  How many persons are in this room, and this room?"

After the small group activity, a whole class discussion should close the lesson.  Questions should be asked by the teacher:
   1.  Can you think of another way?
   2.  Which way do you prefer?  What is your favorite?
   3.  Did you hear an idea or method today you may use next time?  Did you write it in your journal?
        Do it Now.

 This square is an example of vertical, horizontal and diagonal ways of counting to get the same number.

There are numerous square activities where the student does have operational signs.  Some squares have numbers that add up vertically and horizontally with like sums.  There are decimal and fractional squares for third through fifth grades.  I hope to offer all of these on this blog!  We are witnessing internalized visual learning.  It enables deeper thinking processes that promotes memory retention.

Students are again allowed to explain and justify their successful experiences with this simple square.  This participation builds confidence because the learner is taking responsibility for his growing proficiency in number sense.

Problem Solving 



The last four posts have actually been examples of problem solving.  This term expresses many messages.  Most textbook plans have a four step approach.  Read the problem, understand it, decide on a solution and complete it.  Actually, a student is in a problem when there are no strategies or skills within to work it out.  We want our students to assess an exercise, construct patterns and their relationships, then mental images will be formed.  An open discussion with partners or groups is also necessary to listen to their own thinking. They also need others input or challenges.  That is why we start with the very young student with these kind of interactive activities. 

Balances

Illustration from ,Coming To Know Number,  G. Wheatley, 1999

Balance Scales

Classroom students are accustomed to doing addition and subtraction algorithims in a vertical method or a horizontal method.  This is perceived as a operational procedure and lacks purposeful math operations.  When students see 5 + 6 = __, they recognize this as "put in the answer".  This is lacking in depth to their math experience.  It is more of a rote activity.  The optimum way for the students to grasp math ideas is in a setting where they can make sense of their activities instead of just following sets of rules.

Balance scales have been proven to be interesting and worthwhile to enable students to learn addition and subtraction.  Balance is an innate experience.  Children know that balance is needed to walk, ride a scooter, ride a bike, walk on a balance beam on the playground and especially when sharing.  We have to be sure that there is a balance of portions so it is fair to all.

If a teacher uses the balance scales, there is no operational sign, and no mechanical responses.  It provides a visual image that commands a childs mind to think about what is there and what is missing.  The student must examine the scales and process how to act.  If your classroom has a pan balance, it is useful for students to manipulate blocks on both sides to create balance.  This is not mandatory.

Any variety of problem and operation can be achieved on the balance scales.  For our purposes, we will show examples that can be used with older first and second graders.  When we start to put combinations of numbers together (addition) or deconstruction of numbers (subtraction) the balance scales are a meaningful example.

Illustration from, Coming To Know Number, G. Wheatley, 1999

The balances scale at the top of the page illustrates an addition operation. (e.g. 7 = 3 + __ ).  The 3 and the 4 can be be reversed.  This is the communicative property of addition.  That is for third grade.  We could pose the question, "What could we put with 3 to balance 7?  The larger numbers can be on the left or on the right.   As the example at the top of the page, we can pose this scale activity as an addition or subtraction problem. 


The balances scale above can be considered an addition operation. (e.g. 7 = 3 + __ ).  The 3 and the 4 can be be reversed.  This is the communicative property of addition.  That is for third grade.  We could pose the question, "What could we put with 3 to balance 7?  The larger numbers can be on the left or on the right.

I have used the balance scales to teach elementary math in grades first through fifth.  As soon as we start to put combinations of numbers together to find sums and differences, I found that the scales brought out a very attentive and engaging classroom.  What was this, what did it mean, what are we supposed to do?  What are we looking at?  It is a set of balance scales.  They are balanced.  We do have some clues.  What are the clues?  What do we see on the left side of the scales?  What is on the right side?  Is there a value missing?  How can we identify that value?  Students have a visual method of learning.  They have been learning in a similar way with tens frames.  Let them continue this strategy to "move the dots around" mentally or bring out the frames physically to continue looking at numbers collectively.

As the students get more familiar, let them tackle this activity with a partner.  The strength in this lesson is for the student to explain their solution and proof to the class.  If students do not agree, the real learning begins.  This interactive approach increases strategy methods learned from each other.  The students defend their findings and take responsibility for their own perceptions.

Source material:  Coming to Know Number, Grayson Wheatley, 1999

Research One Hundred

Since our number system is based on ten and the powers of ten, counting to one hundred is an awesome feat for the primary learner.  They soon hear the words, thousands, and ten thousands.  Realizing one hundred is complex and needs to be addressed in a constructual way.  By first grade, the students are learning ones, tens and a single unit of one hundred.  Fives and tens can become a mental milepost.

Young learners must learn to know numbers from one to one hundred by intellectually sensing the milestones of 10, 20, 30, 40, 50, 60, 70, 80, 90.  A second grader may have little connection to the number 54.  It could be just a number in counting.  It may be a collection of 54 objects.  He may understand that it comprises five tens and four ones.  The real question is how is it related to 44 and 64?  Unless these students can relate that 44 is ten less than 54 and 64 is ten more than 54, they have not constructed the number in a deeper sense.

The hundreds board is a great tool with so many possibilities.  It help students coordinate tens and ones, two-digit numbers begin to take on meaning in relation to other numbers.  Students can visually examine that 15 is above 25 and is 10 more.  It takes practice.  We of course start at the far right at 10.  This is so familiar because the students are accustomed to counting by tens to one hundred.   After explanation and practice, the young learners will see that every step down is ten more.  Every step up is ten less, wherever you land on the hundreds board.

This illustration is from Coming to Know Know Number, Grayson Wheatley

One game I keep for first graders is hundreds board puzzles.  The small hundreds board is cut up into puzzle pieces.  The students work with a partner to put the pieces together to properly reconstruct the board.  I always have a hundreds board on display in the classroom.  Using a blank hundreds board will help the students move from counting on to constructing an abstract number model with patterns and their relationships.

As a class warm-up, you can place a nickel or penny somewhere on a blank hundreds board transparency.  Ask students what number belongs in this spot?  Have the students share their strategies in their decision.
Repeat this activity several times a week.

Coming to Know Number, G.Wheatley, 1999

Take a blank hundreds board and cut it up into irregular pieces as seen above.  Put a random number in one of the boxes.  (Make sure that it logically fits on the hundreds board)   For convenience sake, laminate these pieces to use with white board markers for re-use.  This another abstract collection created by the student that empowers deeper learning.  It forces the imagery of patterns and their relationships.

The hundreds board is an excellent tool to reinforce collections of grouping or adding.  It can also be a tool for deconstructing or subtracting.  Teachers use Doubles (5 + 5 or 6+ 6), Turnarounds, (4 + 5 or 5 + 4).  Also, there is Doubles Plus One, (5+ 5 = 10 so 5 + 6 = one more, 11.  The pattern relationships continue with counting by twos, fives, tens (count by tens irregularly with numbers like 6 or 9 instead of 2, 5 or 10 that we automatically go to.  This makes the hundreds board a whole unit made up of single parts of one.

Teachers are able to stretch these lessons to " less than" and "more than".  What comes before, what comes after?  At the top of this post the hundred board if highlighted to show odd and even numbers.

One way I taught odd/even is the following.  If the number is even, say, 6, pretend this is six friends.  Do all the friends have a partner ( or buddy)?  The students would start out drawing six circles and draw lines to see.  Yes, everyone was connected to a partner.  Then we have an even number.  No one is left out!  Somehow this really resonated with the young students and they wanted to repeat it to the classroom teacher and classmates.

  Over time and practice, I explained that we only had to concern ourselves with the numerals 0,1,2,3,4,5,6,7,8,9 in seeking out odd and even numbers.  Very quickly, the early learners understood how to master this activity.  I later extended this lesson to explore that even in two digit and three digit numbers, it is still simple to find the odd and even number.  I directed them to underline the number in the ones place (repeatedly).  This number in the ones place determines whether the whole number is odd or even.

Blocks, Dots and Chips

From the nineties on, the American classrooms have slowly included in their math programs sets of manipulitives to accompany math texts and practice books.  Manipultives by themselves are not the key to success for our students.  Without careful teacher planning with a desired goal, manipulitives could in fact impede mathematical knowledge.  The big picture is to allow blocks, bingo chips and other tangible items to be used in a way that creates a visual image in a collection or grouping to substitute rote counting each time for mathematical learning..

The consideration is not if manipulitives are used but how they are used.  The emphasis is always on what is happening mentally during these experiences.  Clear goals need to be set by the teacher so the students understand why maniputives are being used.

  One example I can offer is "Count by Fives"  In a first grade class, I explained that we were going to practice counting by fives to fifty.  First, we used two different colors of interlocking small blocks in groups.  I had a small group of four students.  We used red and blue blocks.  As soon as five blocks were connected, the next group was measured beside the previous set.  Without counting, we could see that collection had equal value.  The children furiously made these sets saying, "Look how many we have!".  I soon had to stop them.  We connected the sets of five across the floor.  Five blue, five red, until we had a long train of blocks.  I then asked them how many blocks were in each set. They were very aware that the number was five.  I said, "O.K., then now we can count by fives"  We begin to skip count by fives.  One child told me that when we got to a ten number, we landed on a set of red blocks. That meant the blues were the five groups.  We eventually made enough of these sets for each of the students to take one back to their individual classrooms.  The train turned into a tower that was displayed at the front of their room.  Each student showed their classmates the tower and demonstrated what it was. They explained the process of the collection by five. Their discoveries when counting and arranging the blocks to fifty enabled them to became the teacher.  The student created their own teaching tool.  There were no numbers on the blocks so they had to visualize.  This became a mental experience.

In the early grades, the math teacher has to continually reinforce the concept of the base ten number system.  Teaching revolves around the focus of the collection of ten as well as what makes up the unit; ten ones.  A geometric setting is another place to construct an abstract example of ten as a unit as well as the ones that dwell within it.  By second grade, students are very familiar with pattern blocks.  They are geometric shapes, specific colors and used as early as pre-kindergarten.  Pattern blocks have different shapes, different number of sides, points or vertices, they stack and make patterns.  As we continue to work with them we find out that the smaller ones in equal numbers will fit on top of the larger blocks.  This becomes an interesting find. In the following activity, each small group has a large quantity of pattern blocks.

In this task, the teacher wants the students to construct ten as an abstract unit. She asks her students:

   1.  Find 10 green triangles.  We are going to make
        a collection of 10.
   2.  What other pattern blocks can you cover using
        using all 10 green triangles?  After your cover
        the larger shapes, push them together making a
        brand new shape.
   3.  Predict:  Will everyone's shape look the same
                                                                                            when we complete making our collection of 10

At the end of the activity, the students were allowed to duplicate their collection on the overhead with transparent pattern block shapes.  Remind the students that they have constructed a collection of ten, made up off ten units of green triangles.  The teacher has again used mental imagery to empower mathematical reasoning and knowledge.The students have also used their spatial sense that further facilitates a richer math learning experience.

Using Mental Images

 Is there a better way to help students internalize number combinations and differences than counting-on or counting back?  Do efforts to have students memorize math facts (addition and subtraction) remain successful over time?  Even in middle school, math students can still be observed to be saying, 9, 10, 11, 12, 13, 14.  Number construction has not formulated.  The memorization method has grown stale.  They continue to use counting to find sums and differences.

Students that have been introduced to thinking in collections have better thinking strategies and flexibility in solving math facts of addition and subtraction.  They can quickly visualize situations and construct in their own minds.

Counting is a central math activity for a child when entering school.  They need to see the patters of of sequence of numbers.  As these counting patterns strengthen, we can begin as early as first grade to introduce collections of dot sets.  This is an abstract concept of number when he or she has mental imagery associated with numerals.  This concept has to be practiced with the students daily with dot cards, tens frames, dominoes or your own made up collection.  It should be a consistent tool that you use.  The tens frame is probably the best by first grade.  The "Quick Draw" process at the overhead for two seconds with dot cards or ten frames will increasingly encourage and challenge students interest to see collections instead of counting the the dots.

If students are allowed to use the visuals to make construct visual addition and subtraction solutions it is superior to memorizing facts.  This takes the drill and timed test out student learning.  Wirtz (1980) observed that children enjoyed 'pushing the dots around in their heads'.

The teacher is demonstrating addition of 8 + 7 before and after using dot patterns.

Using collections, young students are encouraged to create an abstract model or concept of number without  resorting to the procedure the find the number each time by counting.  Encouraging counting each time can  actually interfere with the construction of number.  That is, because it is a rote operation and may be devoid of any mathematical meaning.  We cannot say that counting in itself is not meaningful but that alone may be a substitute for sense making.  (Wheatley, 1999)

This teacher wants the student to look at the dots and move five above in the tens frame  (top) to  the bottom  tens frame.  When complete, there will be one filled tens frame (10) and a tens frame with one 2 dots.The student should be able to visualize the sum of 12 without counting on,  memorizing or finger counting.

To conclude:  Students and teachers will and should always count in the beginning grades in order to establish numbers and their recognition and patterns in the base ten number system.  The point to be made is that by the time students are beginning to combine and deconstruct numbers, dot patterns can be introduced.
Tasks that encourage students to think in collections provide rich opportunities  to create mathematical relationships and become powerful problem solvers.  Using collections rather that counting is a more effective way to encourage students to construct mathematical units.

I have personally used tens frames and dot cards with my young and primary students.  We have used them to construct problem solving examples.  This is exciting for them and although we will have to convert to the numerical algorithms after we learn the abstract way, they already have a visual concept to rely on.  These activities only foster self-reliance, confidence and a sense of mastery.  It enables the student to create with their hands and mind, a solution without relying on counting each time.  It actually helps the students remember their facts. Have blank tens frames and dots that can be purchased (multiple colors) to allow the students to make their own.  Have them create their own frames individually one to ten.  Give them a baggy for storage in their math area.  The possibilities are endless.  They can partner and problem solve.  You can also do the same thing with blank tens frames and bingo counters.  This way you can constantly change the values when you problem solve individually.  My favorite manipulitive is the pictured below.  It is tens frame that has magnetic dots to interact with the teacher, Wow!

These are called magnetic answer boards.

Know Numbers In Groups

 

As we have earlier discussed, many educators have viewed mathematics as acquiring knowledge.  This is done by possibly "stunting" development by way of  memorizing facts and processes.  A different way of viewing math is by building patterns and relationships.  This removes "a set of rules" to something a person can do with materials.  Knowledge is created by solving problems instead of acquired by of seeing and saying.

We've learned that students retain knowledge best when they manipulate, process and work with others during problem solving in mathematics.  Learning materials and manipulitives have a great impact on the students' interest in participation.  The following is a list of questions you should ask yourself before you decide on your "hands on" materials. This research has been tested by Grayson Wheatley, University of Florida.  Some source material is from his book Coming to Know Number, 1999

 - Are the goals dealing with the concepts rather than the procedures?

 - Are the BIG IDEAS being emphasized and made apparent?

- Can the students make sense of what they are learning?  Can they become responsible for their learning?

- Do these activities enhance mathematics learning?

Dr. Wheatley has several examples that enable students to envision thinking in a collection.  He uses this with dot patterns, arrays and dice.                         

A warm up game that can be used in every strand of math is called "Quick Draw"  There is a book with this title.  This photo on the left could be used for this purpose.  The photograph is laid down and covered up on the overhead.  After you have played the gamed before, all you have to say is "Quick Draw"!  The students will automatically come to attention and get ready to draw what is to be revealed.  When everyone is ready, turn on the overhead light for only two seconds.  Turn off, no talking, and students will draw what they saw.  When the heads come up, discuss what they viewed.  You'll be shocked how well your students will become at momentary viewing.  You can see that this grouping strategy with dots on a page can be ordered  in a group or disordered.  The best results is that the students soon will be able to look, momentarily, and know the number without a finger count.

                                                                                    

  Ultimately, our desire is to move the students into the tens frame.  We can show this first with the dot cards and later with the frame. It will show five connected to five.  The counting process will come so naturally as a mental process if one, two or more are missing.  With the very young students, we stay with the dots.  Ultimately, our desire is to move the students into the tens frame.  We can show this first with the dot cards and later with the frame. It will show five connected to five.  The counting process will come so naturally as a mental process if one, two or more are missing.  With the young students, we stay with the dot cards until proficiency.

 
My idea was to use dominoes to connect only those that equaled ten.  My "write-on" dice can also be used  as well as regular dice   Be sure to keep tallies for those that roll a ten! Since our number system is base ten, this is a vital foundation.