Mathematics Grade 6
Strand: EXPRESSIONS AND EQUATIONS (6.EE) Standard 6.EE.9
Students will learn to identify the relationship between patterns and functions.
Invitation to Learn
The Ins and Outs of Functions
The Fly on the Ceiling, by Julie Glass; ISBN 0679886079
There is a powerful pattern identified when any number is put into an equation and consistently follows the rule. This is called a function. When the rule is identified, each number does not have to be solved, but one could simply skip to the input number desired, insert it into the "rule" or equation, and the answer (output) will be given. Functions are easily shown in tables, such as the example below:
It is easy to see that the output increases by one each time. The relationship between the input and output is the key, however. The "rule" is to add 2. If 2 is added to the input number 1, the answer is 3. Therefore, an equation can be formed. An equation is a mathematical sentence that contains an equal sign. The equation for the above example is x + 2 = y. Students should become proficient at spotting the pattern, recognizing the rule, and creating an equation from that rule. The rule or equation should be a one- or two-step problem, or it becomes really difficult to solve.
Another skill that students need to master is the ability to change an equation back into a function table. If the equation is x + 2 =y, the student can choose any number to represent the input (x). They will then "plug in" that number to get the output (y). So, if 8 were chosen for the input, then 10 would be the output. Keep in mind that any variable can and should be used, not just x and y each time.
Moving into the final skill, students need to first be able to graph a function table. This is a simple plotting exercise (taught in 5th grade and in 6th grade Standard III Objective 2). If x is 1 and y is 3, the coordinates will be (1,3). At least 2 (preferably 3) coordinates must be plotted, then connected to create a line (for these types of equations, a straight line will be created). The goal is for students to be able to graph an equation. In summary, here are the steps:
Invitation to Learn
Show students ingredients for cookies. Ask students if they would like to eat each ingredient. They may want the sugar, but not the salt, etc. Explain to students that these ingredients go through a "magical" change from their separate ingredients until they are spit out of a factory machine. The magic, of course, is the mixing of the ingredients and chemical change when they are cooked together. Tell students that today they will be putting numbers through a machine, which will "magically" change the number. The magic, of course, is the function rule. You may give the students a cookie, notifying them that this cookie may stimulate their brain and make them even better mathematicians.
NOTE: You may want to allow students to choose to use a table, rule, or equation, but eventually move students to using equations.
The Ins and Outs of Functions
2x + 3 = y
If x = 1, then y = 5
If x = 2, then y = 7
If x = 3, then y = 9
Then, they need to plot a function table.
Step 1: Students will put an equation into a function table (at least 3 sets to plot).
7 + x = y
Step 2: Students will graph the above table using play dough "dots."
Step 3: They will put the dried spaghetti in all 3 dots. This will ensure that the points are straight.
Step 4: They will repeat the process with the rest of the equations in the set.
Step 5: If done correctly, the 3 lines will intersect with at least one other line on the same graph.
Equation sets to use:
X + 1=Y
X -- 2=Y
2X + 3=Y
X + 3=Y
3X -- 2=Y
X ÷ 2=Y
X - 1=Y
X ÷ 3 + 1=Y
5X - 3=Y
3X ÷ 2=Y
3X + 7=Y
X -- 7=Y
6X ÷ 3=Y
The equations used for the spaghetti graphs were positive slopes (lines that go from right to left). Your advanced learners can be exposed to negative slopes, which are lines that go from left to right. If the equation has both a negative number before X and the second number, it will be a negative slope. For example, -5X -- 3=Y is a negative slope.
0-2 correct: Intervention-These students need direct reteaching instruction
3-4 correct: Practice-These students need extra practice
5-6 correct: Proficient-These students have mastered the content. Give them an enrichment/extension activity to do
Cwikla, J. (2004). Less experienced mathematics teachers report what is wrong with their professional support system. Teachers & Teaching, 10(2), 181-197.
When less-experienced mathematics teachers interviewed, they expressed disappointment that many of their more experienced colleagues lacked content knowledge. Overall, they were not satisfied with the mentoring or collaboration offered by fellow teachers because they often knew more content than their more experienced peers.
Holly, K. R. (1997). Patterns and functions. Teaching Children Mathematics, 3, 312-313.
This article gives many ideas and activities for teaching patterns and functions in elementary grades K-6. Venn diagrams, function machines, and building cubes are some ideas presented.