1. Formulas

2. What is a formula?
A formula is a special type of equation that links two or more physical variables.
Formulas say something about the real world.
Here is a formula that applies to rectangles: l
P = 2(l + w)
The letters represent features of a rectangle.
P stands for the perimeter of a rectangle.
Teacher notes
The letter symbols in formulas are called variables because their values can vary.
In an equation, the letters can take the value of any number that makes the equation true. In a formula, the letters are limited to values determined by the interpretation of the formula. For example, in the formula for the perimeter of a rectangle on this slide, the variables all represent lengths so they can only take positive values. If a variable represented a number of objects, then it can only take whole number values.
l stands for the length of the rectangle. w
w stands for the rectangle’s width. 2

3. Substituting values
We can use this formula to work out the perimeter of any rectangle given its length and width.
P = 2(l + w)
We do this by substituting known values into the formula.
Brittney wants a fence to keep dogs off the lawn.
The lawn is a rectangle 4 m wide and 7 m long.
How much fence does she need?
P = 2(l + w)
Photo credit: © rSnapshotPhotos, Shutterstock.com 2012
P = 2(7 + 4)
Write 7 instead of l
and 4 instead of w.
P = 2 × 11
P = 22 m 3

4. What would the formula be if each bag cost $2.50?
Writing formulas
There is not always a ready-made formula to apply to a problem. Sometimes we need to write a formula.
For example, Stacey buys some bags of chips from a store. Each bag of chips costs $3.
Write n for the number of bags.
Write c for the total cost of the chips.
Write a formula linking n and c:
Teacher notes
Students can test the formula by substituting values in. For example, we know 1 bag of chips costs $3, so putting n = 1 should give c = 3. We can also work out easily that 2 bags of chips costs $6, so check that n = 2 gives c = 6.
Note that the variable n can only take positive integer values.
Mathematical Practices
2) Reason abstractly and quantitatively.
Students should be able to decontextualize the mathematical relationship between the total cost and number of bags purchased and represent it symbolically as a formula, and to recontextualize the formula produced to adjust it when the situation is altered.
4) Model with mathematics.
Students should be able to apply mathematics to everyday life, identifying important quantities and describing their relationship using a formula.
6) Attend to precision.
Students should recognize the importance of giving clear definitions of the symbols they use when they write formulas. A formula is meaningless unless the symbols are defined.
Photo credit: © photomak, Shutterstock.com 2012
c = 3n
What would the formula be if each bag cost $2.50?
c = 2.5n

5. Substitute 105 into the formula instead of n.
Calculating costs
A window cleaner charges $10 for travel plus $7 for every window that he cleans. Write a formula to find the total cost, c, when n windows are cleaned.
c = 7n + 10
Using this formula, how much would it cost to clean all 105 windows of Formula Tower?
c = 7 ×
Substitute 105 into the formula instead of n.
Mathematical Practices
2) Reason abstractly and quantitatively.
Students should be able to decontextualize the mathematical relationship between the amount the window cleaner charges and the number of windows he cleans and represent it symbolically as a formula. They should be able to solve the formula they have generated by manipulating the symbols as if they have a life of their own, without attending to their referents.
4) Model with mathematics.
Students should be able to apply mathematics to solve the window cleaning problem, identifying the important quantities (e.g. travel charge, cost per window) and describing their relationship using a formula.
Photo credit: © Vereshchagin Dmitry, Shutterstock.com 2012 =
= 745
It will cost $745.

6. Matchstick pattern Look at this pattern made from matchsticks: pattern
number, n 1 2 3 4
number of
matches, m 3 5 7 9
Teacher notes
To figure out the formula for the number of matches, m, in pattern number n, look at m and n at each stage. Ask students to think about what they can do to 1 to get 3, and whether they can do the same thing to 2 to get 5. They should find that the rule that works for all these pairs of numbers is multiplying by 2 and adding 1. Some students may find it easier to figure out the formula by looking directly at the pattern, and noticing that the pattern starts with 3 matches and then adds 2 every time.
Mathematical Practices
8) Look for and express regularity in repeated reasoning.
Students should notice that the matchstick pattern is produced by repeatedly adding two matchsticks in the same formation to the existing pattern. Noticing this regularity should lead them to realize that the number of matches can be predicted from the pattern number, the position of that particular pattern in the pattern sequence. They should then be able to construct a general formula linking these values.
What is the formula for the number of matches, m, in pattern number n?
m = 2n + 1

7. Writing mathematical rules
Formulas can describe mathematical rules.
For example, Julio is investigating patterns in the numbers on a 25 square grid numbered along the rows.
He looks at arrangements of numbers in two-by-two squares, like the shaded block shown.
Julio notices that the sum of the numbers in a two by two square is always equal to four times the number in the top left-hand square plus 12.
Teacher notes
Get students to test Julio’s formula on different two-by-two squares. The activity on the next slide contains more puzzles like this that students can try to find formulas for themselves. Students need to be able to generate formulas to express mathematical rules that apply to numbers as well as from word descriptions of real-life situations.
Mathematical Practices
7) Look for and make use of structure.
Students should notice that the arrangement of numbers in the grid means that the value of a number in a certain position can be predicted. For example, the number in any square is always 1 less than the number to the right and 5 less than the number directly below it. They should make use of this pattern to predict the sum of any block using only one of the numbers in that block.
He writes this as a formula:
s = 4a + 12