Algebraic topics at this level should be integrated with the development of other mathematical content to enable students to recognize that algebra is not a separate branch of mathematics. Students must understand that it is an expansion of the arithmetic and geometry they have already experienced and a tool to help them describe situations and solve problems.

Students should use algebraic concepts to investigate situations and solve interesting problems. There should be numerous opportunities for collaborative work. Since algebra is the language for describing patterns, students should have regular and consistent opportunities to discuss and explain their use of these concepts. They should write generalizations of situations in words as well as in symbols. To provide such opportunities, the activities should move from a concrete situation or representation to a more abstract setting. Students at this level can begin using standard algebraic notation. This should be developed gradually, moving them from the previous symbols in such a way that they can appreciate the power and elegance of the new notation.

Students need to learn how variables are different from numbers (a variable can represent many numbers simultaneously, it has no place value, it can be selected arbitrarily) and how they are different from words (variables can be defined in any way we want and can be changed without affecting the values they represent). Students need to see variables (letters) used as names for numbers or other objects, as unknown numbers in equations, as a range of unknown values in inequalities, as generalizations in pattern rules or formulas, and as characteristics to be graphed (independent and dependent variables).

An algebraic expression is a description of some operation involving variables, such as 2b, 3x - 2, or c - d. Expressions involve both numbers and variables, which sometimes follow the same rules and sometimes follow different ones. For example, 2b equals 6 when b = 3 (not 23!). In fifth and sixth grade, students should begin to become familiar with the common notational shortcut of omitting the operation sign for multiplication.

Students in grades 5 and 6 should focus on understanding the role of the equals sign. Because it is so often used to signal the answer in arithmetic, students may view it as a kind of operation sign - a "write the answer" sign. They need to come to see its role as a relation sign, balancing two equal quantities. Students should develop the ability to solve simple linear equations using manipulatives and informal methods. If provided the appropriate background, students at grades 5 and 6 have the ability to find the solution, such as 7 for x+5=12, whether they use manipulatives, a graph, or any other method. It is imperative that when discussions about methods are conducted, many solution methods are described.

Students in grades 5 and 6 should use concrete materials, such as algebra tiles, to help them develop a visual, geometric understanding of algebraic concepts. For example, students can represent the expression 3x - 2 by using three strips and two units. They should make graphs on a rectangular coordinate system from data tables, analyze the shape of the graphs, and make predictions based on the graphs. Students should have opportunities to plot points, lines, geometric shapes, and pictures. They should use variables to generalize the formulas they develop in studying geometry (e.g., p = 4s for a square or A = l x w for a rectangle). Students should be able to describe movements of objects in the plane through horizontal and vertical slides (translations). They should occasionally be able to experiment with probes which generate the graphs of experimental data on computers or graphing calculators. The majority of this work will be with graphs that are straight lines (linear functions), but students should have some experience seeing other shapes of graphs as well.

Building upon K-4 expectations, experiences in grades 5-6 will be such that all students:

E. understand and use literal variables, expressions, equations, and inequalities.

• Students let the length of one side of a square be one unit. They then find the perimeter of one square, two squares connected along an edge, three squares connected along their edges, and so forth, as shown below.
  

  They make a table of values and use it to determine a function rule which describes the pattern. They understand the rule P = 2 x s + 2, and they predict the perimeter of ten squares.

• Students write a LOGO program to draw a rectangle of any dimensions using the variables :LENGTH and :WIDTH.
• In a health unit, students are studying the dietary needs for maintaining healthy bodies. The teacher gives guidelines for the most fat content and cholesterol they should have at any meal. Different groups take different meals and choose two foods at that meal. They determine the fat content per unit of each of the foods and the cholesterol amount per unit and make inequalities which relate these unit values, the number of units, and the maximum amount. The teacher uses a function graphing computer program or graphing calculators to identify the different amounts of each they could have and still be less than the maximum allowable.
• Students match strips with phrases such as "the product of 4 and 5 more than the price of tapes." with symbolic expressions such as 4(p+5).

• Each group of students is given a Mr. or Mrs. Grasshead (i.e., a sock filled with dirt and grass seed which sits in a dish of water). They create a name for their grasshead and begin a diary, recording the number of days that have passed and the height of the grass. At the end of specified time periods, they discuss the changes in the height, the average rate of change over the time period, and the overall behavior of the grass growth. Each group makes a graph of height versus the number of days. The students note whether the graph is close to a straight line.
• Students find the number of tiles around the border of a floor 10 tiles long and 10 tiles wide by looking at smaller square floors, making a table, identifying a pattern, and generalizing their work. They then develop a rule for finding the number of border tiles for an n x n floor.
• Students use play money to act out the following situation and solve the problem.

  A man wishes to purchase a pair of slippers marked $5. He gives the shoe salesman a $20 bill. The salesman does not have change for the bill so he goes to the pharmacist next door and gets a $10 and two $5 bills. He gives the customer his change and the man leaves. The pharmacist enters shortly after and complains the $20 was counterfeit. The shoe salesman gives her a $20 and gives the counterfeit bill to the FBI. How much did the shoe salesman lose?

• Students place 8 two-color chips in a paper cup and toss them ten times, recording the number of red and yellow sides showing on each toss. For each red chip that shows, they lose $1. For each yellow, they win $1. For each toss, the students write a number sentence that shows their win or loss for that toss. Afterwards, the students look for patterns in the number sentences that they have written. They discuss these patterns and then write about them in their notebooks.

• Students use number lines to demonstrate addition of integers. They point at the amount currently in the bank and then slide their finger in the direction (right for deposits and left for withdrawals) over the distance indicated by the amount. As they slide their finger, they keep track of their movements using arrows over the number line, and the teacher keeps track of the operations being done using integers. Through this dual representation, students begin to understand the relationship between the addition of integers and such slides.
• Students are given a variety of objects whose dimensions they must determine. They are given a number line marked from 5 to 27 which simulates a broken ruler. Students work in pairs to develop a process for determining the lengths (to the nearest unit). After they have a workable method and have written an explanation of it in their journal, the teacher replaces the tape with another which is marked from -10 to 10. The students repeat their effort. This process begins to develop an understanding of subtraction of integers, the relationship between distance and the operation of subtraction, and provides an opportunity to show that the answers cannot be negative regardless of the order in which they are subtracted.
• Addition and subtraction of signed numbers is explored using two-colored disks and a number line. Red is used to represent positive numbers and yellow is used to represent negative numbers. When given a problem such as -3 + 5 the students place 3 yellows and 5 reds on the table. They pair up as many red and yellow disks as they can and remove them from the table. In the case of the example, 3 red and yellow disks would be paired and removed, leaving 2 red disks which represents +2.

• Students use tables or two-color chips to help them solve the following problem. A classroom has 25 lockers. The first person to enter the room opened every locker. The second person started with the second locker and closed every other one (the multiples of two). The third person started with the third locker and changed every third locker's state (open to closed and closed to open). This continued until 25 people had passed through. Which lockers would be open?
• Students are given a plastic rectangular shape which they place on the overhead. One group is responsible for determining the relationship between the distance from the screen and the image size (length). A second group is responsible for studying the relationship between the distance from the screen and the area of the image size. The length of the image, the width of the image, and the distance from the screen to the overhead are measured. Then the overhead is moved and the process is repeated so that measurements are taken at six to ten different distances. Each group makes a scatterplot of their data and eyeballs a line of best fit using a piece of spaghetti. They then use the graph to answer questions about the relationship between distance and size or area. They also develop a summary statement describing the relationship.

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• Strings of length 64, 32, 16, and 8 cm with a washer at one end are connected to a screweye or ruler. The strings are swung from a constant height and the number of swings in 30 seconds is recorded. A graph is made plotting the number of swings against the string length. Students study the results and determine if there might be a pattern they could continue. They attempt to answer questions such as will the number of swings ever reach zero? (NOTE: This activity is a good one to repeat at later grades since it appears linear but when very short lengths and very long lengths are used, it becomes clear that it is actually a quadratic relationship.)
• Students are given the times of the Olympic 100-meter freestyle winners both in the men's event and the women's event. Using different colors for the two genders, they produce a scatterplot and use a piece of spaghetti to eyeball a line of best fit for each set of data. They use their lines to determine times in the years not given (when no Olympics were held) and to predict times in the years beyond those they were given. They also determine if the data supports the assertion that the women will some day swim as fast as the men and predict from their lines when that would happen.

• Students are paired to play a game similar to battleship where they attempt to determine where the two lines their opponent has drawn intersect. Both students draw axes which go from -10 to 10 in both the x- and y-directions. They sit so that neither can see the other's paper. The first player draws two lines which intersect at a point with integer coordinates and colors the four regions different colors. The second player gives a coordinate of a point. The first player responds with the color of the region or that it is on a line (or that it is the point!). The second player keeps a record of his guesses on his axes and continues in this manner until the point is determined.
• Students keep track of the high and low temperatures for a month in two different colors on a graph. The horizontal axis represents the day of the month and the vertical axis represents the temperature. At the end of the month, they connect the points making two piecewise linear graphs. They use their graphs to discuss the temperature variations of that month.
• Students consider what happens if they start with two bacteria and the number of bacteria doubles every hour. They make a table showing the number of hours that have passed and the number of bacteria and then plot their results on a coordinate graph.
• Students draw pictures and identify coordinates of critical points. Their partners attempt to create the picture using the coordinates.

• Students use algebra tiles to help them solve an equation. For example, they represent the equation 3x + 2 = 5x by placing three strips and two units on one side of a picture of a balance beam and five strips on the other side. They decide that the balance will stay even if they take the same number of objects off both sides, so they take three strips off both sides and have two units balanced with two strips. Then they decide that 1 unit must balance one strip.
• 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 develop the expression $6.50 X W+$7 as the amount of money in the fund at the end of W weeks. The teacher asks them how much they would like to send to the town, and the students agree on $100. The teacher then asks them to write an equation which would say that the amount of money after W weeks was $100. The students wrote $6.50 X W + $7 = $100. Finally, the students tried a number of different strategies for finding out what W should be. Most of the students used their calculators in a guess-and-check method. Some students went to the computer and used a spreadsheet to generate the amounts for different weeks until it was more than $100. Others expressed the relationship as a composite function using function machines and then used the inverse operations to know to subtract $7 and then divide by $6.50. They used their calculators to carry out the division.

• Using a motion sensor connected to a graphing calculator or computer, the class experiments with generating lines which represent the distance from the sensor against time. They try walking away from the sensor at different rates of speed to determine what effect it has on the line. They start at different distances from the sensor to see the effect that has. They try walking toward the sensor and standing still. Students discuss the relationships between the lines they are generating and the physical activity they do. As an assessment, the teacher has one child walk a line. The students are then asked to write in their journals what someone would have to do to produce a line which was less steep. After closing their journals, the teacher selects people to try their actual processes.
• After measuring several students' heights and the length of the shadows they produce, the data is entered in a spreadsheet, computerized statistics package, or graphing calculator. A scatterplot is formed from the data and the students see that the plot is approximately linear. The technology is used to produce a line of best fit which the class uses to determine heights of unknown objects (such as a flagpole) and the length of the shadow of objects with known heights.

• Students cut out squares from graph paper, recording the length of the side of the square and the number of squares around the border of the square. They look for a pattern that will allow them to predict the number of squares in a border of a 10 x 10 square and then a 100 x 100 square. They describe their pattern in words. The teacher then helps them to develop a formula for finding the number of squares in the border of an n x n square
  (4 x n - 4).
• Students explore patterns involving the sums of the odd integers (1, 1 + 3, 1 + 3 + 5, ...) by using squares to make Ls to represent each number and nesting the Ls. Then they make a table that shows how many Ls are nested and the total number of squares used.

  

  They look for a pattern that will help them predict how many squares will be needed if 10 Ls are nested (i.e., if the first 10 odd numbers are added together). They make a prediction and describe how they found their prediction (e.g., when you added the first 3 odd numbers, it made a square that was three units on a side, so when you add the first 10 odd numbers, it should make a square that is ten units on a side and you will need 100 squares). They share their solution strategies with each other and develop one (or more) formulas that can be used to find the sum of the first n odd numbers (e.g., n x n).

• Students set up a table listing the length of the side of a square (x) and the area of the square (y). Some students use the centimeter blocks to help them find the values. The teacher completes a table of values in a function graphing computer package on the class computer which has an LCD panel for overhead projection or on an overhead version of a graphing calculator. When the students have finished filling in the table, the teacher turns on the overhead and displays her table. The students check their answers and ask questions. The teacher graphs the data on the computer or calculator, and the students use the graph to answer questions such as "If the side was 3.5 cm, what would the area be?" and "If the area was 60 square centimeters, what would the side be?". The teacher uses the trace function to identify the points being discussed.
• Students explore inequality situations such as I have $150. How many more weeks would I need to save my $15 allowance to buy a stereo that costs $200?. They represent the relationship as an inequality, both in words and in symbols, and use play money, base ten blocks, graphs, or trial and error to solve the problem.

• Students keep track of how far they are from home during one specified day. They draw a graph which represents the distance from home against the time of day and write an explanation of their graph in relation to the actual activities on that day.
• Students are presented with a graph representing a student's monthly income from performing lawn care for people over the past year. The graph shows no income during the months of November, December, and March. They write a story which explains the behavior of the graph in terms of the need for services over the year.