Students can develop a strong understanding of algebraic concepts and processes from consistent experiences in classroom activities where a variety of manipulatives and technology are used. The key components of this understanding in algebra, as identified in the K-12 Overview, are: patterns, unknown quantities, properties, functions, modeling real-world situations, evaluating expressions and solving equations and inequalities.

In grades K-2, students use pictures and symbols to represent variables, generalize patterns verbally and visually, and work with properties of operations. Although the formality increases in grades 3 and 4, it is important not to lose the sense of play and the connection to the real world that were present in earlier grades. As much as possible, real experiences should generate situations and data which students attempt to generalize and communicate using ordinary language. Students should explain and justify their reasoning orally to the class and in writing on assessments using ordinary language. When introducing a more formal method of communicating, such as the language of algebra, it is helpful to revisit some of the situations used in previous grades.

Since algebra is the language of patterns, the mathematics curriculum at this level needs to continue to focus on patterns. The use of letters to represent unknown quantities should gradually be introduced as a replacement for pictures and symbols. The use of function machines permits the introduction of letters without the need to move to formal symbolic algebra. Since they have had the opportunity to experience real function machines such as the calculator or a gum bank, where one penny yields two pieces of gum, the notation of function machines should make sense.

Here the box is thought of as the function machine times 2 which takes in a number "a" and produces a number "b" which is twice "a." Students can use such symbols to communicate their generalization of patterns. They put two or more machines together making a composite function; for example, they can follow the times 2 machine with an add 3 machine. They determine not only what each input produces but also what input would produce a given output.

Students should continue to communicate their generalizations of patterns through ordinary language, tables, and concrete materials. Graphs should be introduced as a method for quickly and efficiently representing a pattern or function. They should develop graphs which represent real situations and be able to describe patterns of a situation when shown a graph. For example, when given the graph at the right which shows the relationship between the number of bicycles and wheels in the school yard, they should be able to describe the relationship in words.

Students in grades 3 and 4 should continue to use equations and inequalities to represent real situations. While variables can be introduced through simple equations such as 35 ÷ n = 5, students should be viewing variables as place holders similar to the open boxes and pictures they have already used. At these grade levels, they need not use variables in more complicated situations. Given a situation such as determining the cost of each CD if 5 of them plus $3 tax is $23, they should be permitted to represent it in whatever way they feel comfortable. Students should be able to use, explain, and justify whatever method they wish to solve equations and inequalities. Some may continue to use concrete materials for some situations; they might count out 23 counters, set aside 3 for the tax, and divide the remainder into 5 equal piles of 4. Others might try different numbers until they find one that works. Some students may write 23 - 3 = 20 and 20 ÷ 5 = 4. Still others may want to relate this to function machines and figure out what had to go in for $23 to come out. It is important for students to see the diversity of approaches used and to discuss their interrelationships.

Students should continue to examine the properties of operations and use them whenever they would make their work easier. There are some excellent opportunities for providing a foundation for algebraic concepts in these grades. For example, explaining twodigit multiplication by using the area of a rectangle (see figure below illustrating 13 x 27) provides the student with a foundation for multiplication of binomials, the distributive property, and factoring. While the teacher at this grade level should focus on the development of the multiplication algorithm, the teacher of algebra several years later will be able to build on this experience of the student.

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

1. Understand and represent numerical situations using variables, expressions, and number sentences.

• Students do comparison shopping based on items that are for sale in multiples. For instance: If chewing gum is sold at 3 packs for 85 cents (p = 85/3), is that a better or a worse buy than a single pack for 30 cents (p = 30)? They sort their examples into groups where the multiple buy is a better deal, the same deal, or a worse deal than the single package deal.

• One student has been folding origami cranes to send to Hiroshima for Peace Day in August. He brings the 47 cranes that he has folded so far to class and asks for help to fold many more. The class decides to have each of the 26 students fold one crane each week for the rest of the school year. The teacher asks groups of students to find a way to determine how many cranes will be in the collection after some given number of weeks. She starts off the discussion by having students list the numbers for the first few weeks:

  47,
  47 + 26,
  47 + 26 + 26,
  47 + 26 + 26 + 26,

  and so on.

  They figure out whether they can reach 500 cranes by the end of the year.

2. Represent situations and number patterns with concrete materials, tables, graphs, verbal rules, and number sentences, and translate from one to another.

• Students compare two allowance plans. Plan A provides an allowance of $5 the first week and adds $1 each week. Plan B starts with 1 cent and doubles the allowance each week. Using calculators, students make tables listing the number of the week, the amount of allowance under Plan A and the amount of the allowance under plan B. They complete several rows of the table so that they understand what is happening with each plan and see that Plan B soon overcomes Plan A. They might want to try to use physical objects such as centimeter cubes to demonstrate the behavior of the two plans visually.

• Each student is given an even number of square tiles and asked to use them all to make a rectangle with two columns. Students are asked to notice that the heights of the rectangles are different for different starting numbers of tiles. They collect the data into a table, giving each student's name, the number of tiles used, and the height of the rectangle. They understand that the number of tiles is the area and can figure out the height of the rectangle if they know the number of tiles that are used - that is, they can verbalize that the height of the rectangle is half the total number of tiles.

• Students read 1 Hunter by Pat Hutchins, wherein a determined hunter looks and looks and looks for animals but sees none, even though the reader can clearly see 2 elephants, 3 giraffes, 4 ostriches, and so on, up to 10 parrots. They are asked how many animals in all the hunter was unable to see. Students use graphs, concrete materials, pictures, and number sentences to express their understanding of the situation.

• Students play Guess My Rule by suggesting inputs and having the rule-maker (the teacher or a student) put the corresponding outputs into a table like this one:

  Input	Output
  3	7
  1	4
  16	19
  .	.
  .	.

  Students should always be challenged to show they understand the rule by giving a verbal explanation of it. Partially filled-in Guess My Rule tables are a good assessment technique to evaluate the students' inductive reasoning power and their level of comfort with arithmetic operations.

3. Understand and use properties of operations and numbers.

• When the students are introduced to twodigit by twodigit multiplication, they begin with a problem of finding the area of a rectangular field which is 37 feet by 44 feet. They know they need to multiply the numbers to find the area, but they don't know how to multiply without calculators. The teacher draws a rectangle and uses a line to divide the width into two regions which are 30 feet and 7 feet. She does the same with the length, cutting it into lengths 40 feet and 4 feet. This divides the rectangle into four smaller rectangles (30×40, 30×4, 7×40, 7×4) all of which are multiplications the students can do.

• Lea and Suzanne discovered a method for multiplying even numbers by six easily. Their method, applied to the example 6×24, is:

  Cut the other even number in half	12
  Add a zero	120
  Add the number	120+24=144

  When they told their classmates their discovery, they were stumped when they were asked why it worked. The teacher, grasping the teachable moment, divided the class into groupsand challenged them to do a few examples using the girls' method and try to figure out and explain why it worked.

• Students and teacher together work through Robert Froman's book, The Greatest Guessing Game: A Book about Dividing to reinforce their notions of division.

• Students explain that they solved a problem like 300 - 56 mentally by first subtracting 50 and then subtracting 6, since that is the same as subtracting 56. They also do 25×7×4 by first multiplying 25×4 and then multiplying by 7. Such simplifications will give a good foundation for later work in algebra.

4. Construct and solve open sentences (example: 3 + _ = 7) that describe reallife situations.

• In an assessment situation, groups of students are asked to describe in words the situation of four people sharing a five dollar bill found on the way to school, and then to transform it to symbolic form using pictures, symbols or letters.

• Students want to help the New Jersey environment and raise money at the same time. They discover that in two bordering states (New York and Delaware), plastic soda bottles can each be turned in for a 5 cent refund. They write an equation which represents the amount of money they will receive for b bottles. Students answer questions such as How much money will we get for 25 bottles? and How many bottles will we need to make $10?

• Students are presented with a function machine representing the situation of buying music tapes for $5 each through the mail and paying a $3 shipping and handling charge for the order. They answer questions such as How much would it cost for 5 tapes? and How many tapes were bought if the bill was $43?
  

References

Fromer, Robert. The Greatest Guessing Game: A Book About Dividing. New York, NY: Thomas Y. Crowell Publishers, 1978.

Hutchins, Pat. 1 Hunter. New York, NY: Greenwillow Books, 1982.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.