Making and Using x-y (Input-Output) Tables

Input-Output tables are an excellent way to introduce students to linear, proportional, and non-linear algebraic reasoning. When set up correctly, students can use the tables to describe relationships between the independent and dependent variables; how the output is changing between each row; and what transformations take place between the input and the output.


Questions to Ask about Input-Output Tables

Real world situations

• Unit price (specials, 2 cans/$5)
• Constant speed (time vs. distance traveled)
• Year vs. age

What is the rule inside the machine?

• Is it a one-step or multi-step transformation?
• Is the rule constant or changing?
• Is the rule describable, either in words as y = f(x)?

What is the pattern?

• As x increases, what happens to y?
• How can you predict the next row on the table?
• If you skip a few rows on the table, what would the 100th row be?

Important numbers

• What happens when you plug in 0?
• What do you have to plug in to output 0?
• By how much do the inputs and outputs increment each time?

Translate table to other forms

• Rows into (x, y), graph on coordinate plane
• Come up with situation or graphic that the table could describe
• Predict what characteristics the graph will have based on the table.

 In this problem, students can make a connection between the term in the sequence and how many squares form that picture. They can see that as the numbers in the Input column increase by 1 each time, the number of squares grows by 2 every time. You can (literally) highlight that change visually by shading the two new squares that are added on to form the new shape. As an extension, you can ask students to analyze other characteristics of the shape by gathering data and making conjectures- for example, finding the perimeter of those shapes, or redrawing it to be a 3-d figure and finding surface area or volume. Or, students may choose to do away with the original independent variable and find the relationship between area and perimeter, or any other two quantities.

The utility of Input-Output tables lies in their flexibility to span grade levels and depths of content, from elementary through calculus and beyond. Students should be able to use tables as a tool to organize and interpret many instances of data and numerical relations.

Solving One-Step Linear Equations, Part 1

This lesson will introduce students to the concept of solving simple equations. The big ideas are:
- The solution to an equation is the number that makes it true.
- You can write a different yet related equation to make the problem simpler.

Step one, bring on the fact families. Remember these?
3+5=8
5+3=8
8-3=5
8-5=3
As long as students have a decent grasp of what addition and subtraction mean, you can use some kind of manipulative to show how the different facts fit together. Using a rectangular array, the same can be done with multiplication and division fact families. When leading students to write out the fact families, ask them to find the patterns about where the largest number is in the addition/multiplication and subtraction/division problems. Is it the part or the whole of the group?

Then, replace one of the numbers in a family with a variable or empty box.
3+[ ]=8
[ ]+3=8
8-3=[ ]
8-[ ]=3
Ask students to fill in the spaces, and then pick out which fact is easiest to "solve" for. They might choose one of the addition facts and solve by counting on, or they'll choose the subtraction fact with all numbers and operations on the left-hand side (LHS) of the equation, and the unknown isolated on the right-hand side (RHS). Whichever one they choose, make sure they check their answer by taking the solution and replacing it in the equation they chose; if it turns out true, then their solution is correct.

Next up, write an equation that has one unknown/blank, such as:
[ ]-6=4
Ask students to write the other three equations that will be in the same fact family (What numbers? What operations? What are the parts and whole?) Circle the equation that has all numbers on one side of the equal sign, and the unknown isolated on the other side. Repeat with similar equations that have the blanks in any of the three spaces until students can identify what different but related equation they would use to solve it.

Resources and Extras:
This worksheet from math-drills.com has algebraic equations with small numbers and the basic operations, with one number blanked out. Have students choose their problems, then write the fact families for them, choose the "easy" equation to solve, then fill in the blank. Other versions of this worksheet have symbols or variables in place of the blank.

Dice are a good tool for getting more tactile with this lesson, especially in the addition/subtraction fact families for lower-skilled students. Roll two dice, place the dice next to each other, and write the addition fact that's shown. Physically change the place of the two dice to show the second addition fact. For example, if they rolled 4 and 5, that gives you 4+5=9 and 5+4=9. Start writing the subtraction fact as 9-[ ]=[ ] and remove one of the dice. Ask, "How many did I take away? How many are leftover? Where do each of those numbers go in the sentence?" Students can practice drawing out each of the problems they do with the dice along with the fact family. Slightly higher-level students can do the same with multiplication/division problems by drawing the array.