Since algebra is the language of patterns, this should continue to be a major part of the mathematics curriculum. Ordinary language should always be a method of communicating which is stressed. Students should explain and justify their generalizations to the class and in writing on assessments. At this level, the use of letters should be gradually introduced as replacements for pictures and symbols. The use of function machines permits the introduction of letters without the need to move to formal symbolic algebra. Given the opportunity to experience real function machines such as the calculator or a gum bank where one penny yields two chicklets, the following symbol should not be confusing.

Students can use such symbols to communicate their generalization of patterns. They put two or more machines together making a composite function. They determine what they need to do if they want to determine what the input (a) was if they are given the output (b).

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 shown the graph on the next page which represents the distance from school for a child's ride home, they should be able to present a scenario which describes the event.

     [Graphic Not Available]

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. There should be no effort to have them use variables in more complicated situations. If faced with 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 however they are most comfortable. Students should be able to use, explain, and justify whatever method they wish to solve equations and inequalities. Some may wish to 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.

Students should continue to examine the properties of operations and use them when they would make their work easier. There are some excellent opportunities for providing a foundation for algebraic concepts in these grades. For example, introducing two-digit multiplication by using the area of a rectangle provides the student with a foundation for multiplication of binomials, the distributive property, and factoring. While the teacher at this grade level should focus only on the development of the multiplication algorithm, the algebra teacher at a future grade level needs to know that this was done so they can build on that experience.

Building upon K-2 expectations, experiences in grades 3-4 will be such that all students:

A. understand and represent numerical situations using variables, expressions, equations, and inequalities.

• Students analyze the following situation:

  Bobbie collects bottle caps. Gregg gave her 31 caps and Albert gave her 27. If B represents the number of bottle caps Bobbie had before the gifts, write an expression indicating how many she has now. If N represents her new amount, write an inequality telling what you know about the number she has now.

  The students use words to describe the relationship and write the expression B + 58 for the amount she has now and N >= 58 for the inequality expressing what they know about the number she has.
• Students want to use their class fund as a donation to a town in Missouri devastated by the summer floods. They decide that everyone in the class is going to contribute 25 cents a week. The fund already contains $7. The students use their calculators to compute that $6.25 will be added each week by the 24 students and the teacher. The teacher then makes a chart on the board having the headings "Number of Weeks" and "Amount." She writes a 0 under the number of weeks and $7 under the amount column. She has the students explain what this represents. The students explain that to find the amount in each week they would add the contributed amount to the previous total. They discuss that the first week would be $6.25 + $7. The teacher writes it exactly that way and then has them compute the total. The second week she writes as $6.25 + $6.25 + $7 and then asks the students to give her a way to write $6.25 + 6.25 as a multiplication. This continues, with the teacher writing $6.25 * 3+$7 and $6.25 * 4 + $7. Then, she asks the students to write an expression which will tell them how much is in the fund at the end of w weeks ($6.25 x w + $7).

• 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 for Plan A and the amount of the allowance for 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 visual the behavior of the two plans.
• 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 rectangles are different for different starting numbers. They collect the data into a table, giving each student's name, the number of tiles used, and the number of rows in the rectangle. They should understand that the number of tiles is the area and then be asked to find a way to figure out the number of rows in the rectangle if they know the number of tiles (e.g., N/2).

• When the students are introduced to two-digit by two-digit 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. The teacher draws a rectangle and draws a line dividing 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. When students seem to understand the area model, the teacher begins to make the method a bit more abstract by having the students first write (30+7)(40+4) and then 30(40+4)+7(40+4). She mentions that the being able to do 30(40+4) is so special that mathematicians have given it the name of distributive property. Even after students can perform 34*47 with a traditional paper-and-pencil algorithm, she brings back the distributive property to show students how it can help them multiply special numbers quickly in their head. For example, 21 x 99 = 21 x (100-1) = 2100 - 21 = 2079.
• Lea and Suzanne discovered a method for multiplying even numbers by six easily. Their method, applied to the example 6 x 24, was:

       Cut the 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. They decided to write a letter to the mathematics supervisor asking her to explain why it worked. The supervisor wrote back explaining that they were really doing 6 x 24 = (5 + 1) x 24 = 5 x 24 + 24 because "cutting the number in half and adding a 0" was really multiplying by 10 and dividing by 2. They received certificates rewarding them for their creativity and the willingness to follow through on a question.

• 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 x 7 x 4 by first multiplying 25 x 4 and then multiplying by 7.

• Students describe in words the situation of four people sharing a five dollar bill found on the way to school and then transform it to symbolic form using pictures, symbols or letters.
• Students want to help the environment and raise money as well. They discover that in bordering states (New York and Delaware), plastic soda bottles can 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 from QVC and paying a $3 shipping and handling charge. They answer questions such as "How much would it cost for 5 tapes?" and "How many tapes were bought if the bill was $43?"