MATH first page

Activities

The following are activities we have used in the Number Sense workshops for training teachers.  They also have been used successfully in our adult classrooms.  If you are interested a particular activity, need clarification or more information,  we Welcome Your Input and Suggestions!  

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Ice Breakers (Opening Activities)

Almost any of the Activities listed below may be used as an opening activity.  However, 1 and 2 listed below are particularly useful while the group is gathering before the workshop begins.

1. Fold the card stock in front of you in half lengthwise. Write your name, where you teach adults , and the level you teach in large letters, on one half of the card. Stand it in front of you, on the table, so all can see.

2. Write one or two words that describe at least one math topic your students experience particular difficulty learning, and therefore, topics you would like addressed in these workshops. Use individual Post-its for each topic. Post the topics on the blackboard.  

During the Feedback period, prioritize the topics by re-arranging the Post-its on the blackboard.

Activities: (Objective b.)

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Common Math Difficulties

The following are common difficulties recorded by teachers during workshops are:

Place Value

Addition/Subtraction

Multiplication/Division

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Defining Addition:  (objective b.)  This activity is particularly helpful as an affective opener.

Instructions given to those entering room. Form groups of three, introduce each other, follow directions. Different tables contain different items: beans, pokers chips, cuisenaire rods, and a variety of objects- all to be counted.

"It is necessary that you know how many chips there are on the table so that you can let you supervisor know. Someone will be coming in to get the information. You are constantly being interrupted, however, so you may not get the job done before that someone gets to you office. That someone may have to pick up the count, or know exactly where you were, that your count was correct, or exactly how many of the objects there are without having to start all over again. You may not use paper, pencil, pen or any other means of written or spoken communication."

As a result of this activity participants demonstrate that addition involves collecting items in groups with which they have something in common.  They also tend to group in meaningful sums, most often fives and tens.  They identify items by color, shape, texture.  Sometimes they give added value to centain items.  Since there can be no written or spoken communication among participants and between participants and presenters, their counting system must be clearly evident to an onlooker.  

Feedback to the large group includes reflection on the groups cognitive processes during the exercise.

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Looking at Numbers (Visualizing Numbers) (Individual Activity with Feedback)

Picture five dots in you minds eye. Then draw a picture of them.  Discuss the picture.  Does it look like die, domino, or playing card?  Are the dots all in a row or three in one row two in another?  How does your picture of the number affect your ability to calculate?  Different ways of  looking at the number can lead to the concept of "part part whole "(3 plus 2 equals 5)  or help with calculations ( 7 is a group of 5 with 2 left over.)  Try another number.

Divergent thinking: What are other ways of thinking about 5 that is not numerical? (Four wheels on a car and a spare; fingers on a hand; two eyes, two ears,and a nose, etc.)?   How about ten?  How do these mind pictures help the identification of "part part whole."

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Anchor Numbers (Large group activity, Brainstorm)

Participants describe the strategies used to add a column of figures, i.e. adding to ten or a favorite number or to number combinations.  Identify numbers commonly used as anchors.  Do adult students use the same anchor numbers in and/or outside the classroom?

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Odd and Even Numbers. (Small Group Activity with Feedback)

Given handful of objects (beans, chips, paper clips, pennies, etc.) each group lines  the items up in pairs.  If the given number of objects is uneven there will be a left over object without a pairing, a visual representation of an odd number.

Combine two different groupings: an even with an even, an odd with an odd, and even and an odd.  Predict the result of combining three or more groupings.  Relate this information to predictions about the sums of numbers.

Participants at each table or in a row, hold hands with a partner. Are there an even or odd number of participants at a table or in a row?  Combine tables or rows. Proceed as above. Discuss the advantages of using whole body activities in a classroom of adults.

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Greater Than, Less Than   (Small Group Activity with Feedback)

In small groups participants address the importance of the concept of greater than or less than when adults are dealing with mathematics problems both in and outside of school.  Using chips, beans, clips, a Hundred Chart, Number line, or participants themselves, etc., each group designs an activity which demonstrates the concept of greater than and less than a given number or an anchor number.  Particular attention should be paid to the concept of less than since this seems to be one which causes adults the most difficulty.  Address the relevance of these concepts to estimating and test taking skills.

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Closer to, Farther From

This activity is similar to that above.  The use of manipulatives for visualizing the concept is particularly meaningful even though it may seem more difficult at first.  The number line and hundred chart are the most obvious means. A human number line is particularly effective a are distances in local geography,  bus or train lines or ideal body weight charts.  Encourage participants to use the less obvious demonstrations.  Participants address this concepts relevance to estimating, and rounding.

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Looking at Number Operations (Part Part Whole)

Defining Number Operations  (Objective d.) (Small group with feedback)

Use  the handout "Different Ways of 'Seeing': Basic Operations in a Problem" (see Materials)

In Groups of two or three, participants write a story problem for a given operation. Discuss which of the definitions of operations their story represents?  When story is written it should create a picture in the mind. How does that picture tie in to a definition of the operation. Do the stories represent all the definitions of the operations? Which ones were not included? Why?  How does limiting ones definitions of each operation affect an adult student's ability to solve story problems?

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Looking at Number Facts (Objective c.d,)

Learning Strategies for Memorizing Number Facts

Addition Strategies

Construct an Addition Table (Individual or small group activity with feedback)

Participants may be presented with a blank addition table, a partially completed table, or a completed table. Constructing a table from scratch allows a participant (or our adult students) to recognize personal strategies used in completing the task. Discuss these strategies and how they may be applied as a means of helping students in memorizing addition facts. (Using the numbers 0 to 15 for the table allows participants and/or adult  students to recognize more readily that the patterns in the sums are present beyond 10 x 10. )

In the interests of time, a "Class Addition Table" may be constructed with groups or individuals responsible for constructing individual tables  

How long does it take for a pattern to be discovered?  Does everyone see the pattern at the same time? What cognitive processes do we observe in our attempts to see patterns?  What are the advantages of working in groups for these activities? 

Looking at the Completed Addition Table

In small groups, using an addition table , identify which sums can be learned using the following strategies: One or Two more or less, Odds and Evens, Doubles and Near Doubles, Sums to Ten and adding Ten to a number, One less than ten, Commutative Property, etc.

Note how few facts are left and therefore how important strategies are to memorization.

Note the identity addend, what happens when evens are added, odds are added, evens added to odds.

Visualize distance to ten using fingers on both hands.  Visualize distance to five using one hand.  When we do that are we using the concept of addition?  How would our adult students answer this question?

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Multiplication Strategies

Construct a Multiplication Table (Individual or small group activity with feedback)

Participants may be presented with a blank table, a partially completed table, or a completed table.  Constructing a table from scratch helps a participant  (or our adult students) recognize that the personal strategies used in the task may be helpful when memorizing multiplication facts.  Discuss these strategies and how they may be applied as a means of helping students in memorizing multiplication facts. (Using the numbers 0 to 15 for the table allows participants and/or adult  students to recognize more readily that the patterns in the products are present beyond 10 x 10.)

In the interests of time, a "Class Multiplication Table" may be constructed with groups or individuals constructing individual tables, i.e. the 9's, 4's, 5's etc.  However, some strategies may not become evident in this instance.

How long does it take for a pattern to be discovered?  Does everyone see the pattern at the same time? What cognitive processes do we observe in our attempts to see patterns?  What are the advantages of working in groups for these activities? 

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Patterns (Small or large group activity with feedback)

In small groups, using a 15 x 15 multiplication table, find the patterns in the following tables: 0's, 1's, 10's. Note the 2 times table and even an odd charactistics of other individual tables.

The 5 Times Table.

Note last digit in (5 times an even number) and (5 times and odd number).  Participants react to the statement , recognize, or discover that 5 x any number = 1/2 that number x ten. Why?

The 9  Times Table.

The sum of the digits of the products is always 9. Looking at the products between 9 and 90 what pattern is seen.  Does it continue past 90?

9 x a number = [(number -1) x 10] + [9 - (number -  1)],  9 x 6 = [(6 - 1) x 10] + [9 - (6-1)],  9 x 6 = (5 x 10) + (9 - 5),  9 x 6 = (50 + 4 )= 54.  Is this a general rule?  How could you prove it?

Other Tables

Using a calculator with a repetitive function key, key in 4 +, =, =, = etc. Note the emerging patterns. How many can you discover?   Can you tell if a given number is a multiple of 4? How would this knowledge help your students?  Try the same activity with another number.  

Look at the 3 and 6 times table.  Look at the sum of the digits in the products.  Is there a pattern?  How can this knowledge help our students?

Look at the "squares"

Participants reflect on personal cognitive process during these activities.  React to the use of a calculator  for these activities in their classrooms.  Why are they advantageous?  How many do you need?

Memorizing Multiplication Facts  (Large group discussion)

Using recognition of the commutative principle in the Multiplication Table and the patterns discovered, identify those multiplication facts that must simply be "memorized" and those for which students may be able to use the patterns or personal strategies to recall.

Identify the products on the Table which appear more than twice.  Which appears most often?  Why?  How can this information help our students?

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Drawing Numbers  (Individual activity with feedback)

Using a blank centimeter grid or graph paper, participants are given numbers on the Multiplication Table  The numbers are then to be drawn on the paper by shading in blocks that form rectangles or squares.  For example 12 may be drawn by shading a rectangle 6 blocks in a row by two blocks  in a row on the paper.  It may also be drawn by shading 12 blocks in one row or 4 blocks in three rows.

Discuss what happens with numbers like 7.  What about 9?  What numbers form squares?  Which have only one representation- a singular column or row?  Which numbers on the Multiplication Table have the more than two different representations?  More than four?  Predict which has the most.

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Looking at Subtraction:

In Groups of two or three, participants write a subtraction story problem.  Discuss which of the definitions of operations their story represents?  When story is written it should create a picture in the mind. How does that picture tie in to a definition of the operation. Do the stories represent all the definitions of the operations? Which ones were not included? Why?  How does limiting ones definitions of each operation affect an adult student's ability to solve story problems?  Feedback to large group.

In groups of two or three, using available materials,  cuisenaire rods,  paper clips, construction paper, beans, pennies, themselves, etc. develop illustrations of the definitions of subtraction.

Play the game of "Giant Steps" to illustrate absolute numbers, and the definitions of subtraction.

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Looking at Division:

In Groups of two or three, participants write a division story problem.  Discuss which of the definitions of operations their story represents?  When story is written it should create a picture in the mind. How does that picture tie in to a definition of the operation. Do the stories collected represent all the definitions of the operations? Which ones were not included? Why?  How does limiting ones definitions of each operation affect an adult student's ability to solve story problems?  Feedback to large group.

Topics for discussion in small groups with feedback to large group:  Why are the following important for adults to know?  A Jigsaw Group Activity Model adapts itself well to this activity.

Using the constant key on a calculator illustrate division as a series of like subtractions

Use the multiplication tables for division facts

In small groups-using beans/water/rice in a jar and a measuring cup, or cuisenaire rods, beans, construction paper, etc. construct illustrations of division problems.  

Place Value and our Number System:

The following activities call for participants, working in small groups, to recall, investigate and/or discover what they know about our place value number system.  Feedback to the large group.  

Ask participants to describe their algorithm for addition. Do they always do it the same way?  Do we ever add columns from left to right?  How about when we estimating the bill at the supermarket?

Use A Calculator to:

Math in Everyday Life:  It's Situational!

In small groups ask participants relate to the following

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Magnitude of Numbers

Many of our adult students have difficulty working with large numbers.  As for ourselves as teachers, we hear about numbers daily and yet we may not have an appreciation for their size even though we acknowledge their existance. To help students appreciate the magnitude of familar large numbers have participants estimate the answers to the following.  Then have them use a calculator or hand calculation to varify their answers:

How long would it take to count to 1,000,000 if you counted each second continuously, 24 hours a day?  How long to count to 1,000,000,000?  

If you spent $1000 a day, how long would it take to use up a million dollars?  How about a billion dollars?  Discuss the importance of recognizing the magnitude of numbers.

The U. S. Government has found it has a 2 billion dollar budget surplus.  If we continued to accrue such a surplus each year and applied it to our national debt of 5 trillion dollars, how long would it take to pay off the debt?  In what way might the knowledge of the magnitude of numbers affect the decisions we make in our daily lives?

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Closure and Next  Step

The participants are asked for their reactions to the workshop and its activities. Participants express their comfort/discomfort level with regard to conducting the above activities.

Participants are asked to form partnerships of two or three, and among themselves select three or more activities illustrating the concepts discussed in workshop.  These may be activities used in the workshop or new ones created by the teachers as a result of participation in the workshop.

Between the two sessions "The Next Step" is to return in a month with the results of these activities or ones they have devised themselves..

At the end of the second session participants are assigned the task of using selected workshop activities or those developed by their peers in their adult classes over the next month.  Their partners may be used as sounding boards or helpmates.  Workshop presenters are also available for reaction, suggestions, and further assistance.