1.1 Numbers and Operations

Worksheet 1.1: Numbers and Operations

Objectives:

• Students should understand that this class involves communicating mathematical reasoning.
• Students should understand that a major component of this class is talking with other students about mathematics.
• Students should begin to develop the concept of covariation.
• Students should review basic rules of fractions.
• Students should review the conventional order of operations.

Suggested Lecture Breaks:

• MWF: You have two days. Get through at least the activity on the first day and as far as possible the second. This section should be mostly a review for them, so rushing through things the second day is okay.
• MW/TR: You have two days. Get through the order of operations lecture and student problems on the first day and finish on the second.

Suggested Lecture Notes:

Syllabus [20 min.]

[15 minutes] Introduce yourself, and go over the syllabus.

Things to highlight from the syllabus:

• Fixed night exams and the policies associated with them
• Online homework through Webwork
• Canvas
• MRC hours
• Structure of the course in comparison to math courses they may have had in high school (i.e. emphasis on group-work, team quizzes, reading assignments, shorter lectures). Tell students that this class will be different in that we expect them to reason about mathematics and talk through their reasoning. Some problems have more than one right answer and some problems have no right answer. In this class, we are not aiming to just get answers, but rather to develop understanding.

[5 minutes] Give students time to introduce themselves (within groups or as a class).

Notes/Suggestions:

Make sure to take attendance.

• Print off extra copies of the first day activities for students who have not yet bought their workbook (at least enough for half the class).

Group Work - Order of Operations Activity [20 min.]

• Introduce the activity by writing $6 \div 2 (1+2)$ on the board.
• Have students work on their own, then encourage them to share their answers. Have someone from each group write their work on the board. If all resulting boardwork is identical, find someone in class with different work and have them share, as well.
• Initiate a class-wide discussion about boardwork. Ask someone with each answer to explain their work. Ask the class how we determine who is correct.
• Someone will say PEMDAS. Refute this idea by asking, "But how do we know this gives the correct order?"
• Give the order of operations (PEMDAS) and reveal that it is simply an order that a group of people agreed upon years ago because it makes the most sense and allows people to be on the same page. An analogy to use is traffic laws. They are also a selected order/set of rules that everyone has agreed upon and if we didn't have them there would be chaos on the roads.
• Bring the class back to their worksheet. Have them do 3 (a) and (b). Bring this together for a discussion on parentheses.

Order of Operations [20 min.]

• Write PEMDAS on the board and give an example, i.e.
  • 6+ \frac{16-4}{2^2+2} - 18 \div(3-6)^2
• Make sure to write ALL steps on the board.
• Be sure to point out that if there are both multiplication and division, then we simply work from left to right. To illustrate this, think back to the example in the group activity. Alternatively, you can use $18 \div 9 \cdot 9$ and point out that this equals 18, NOT 18/81.
• Have students work on Problem 1. Time permitting, ask a few students to put their work on the board.

Primes and Factors [15 min.]

• Give definitions of prime number and factor. Ask the class for a few examples of primes, and write these on the board.
• Give an example where you calculate the prime factorization of a (2 or 3 digit) number.
• Have students work on problem 2.
• Give the definition of the greatest common factor of two numbers.
• Give an example where you find the GCF of two numbers. Do this by listing all factors of the numbers and identifying the largest factor in both lists.

Fractions [25 min.]

• This section is assumed to be a review. Remind students of the definitions of numerator, denominator, and equivalent fractions. Make sure to explain why equivalent fractions are important. Then give an example each of:
  • putting fractions in lowest terms
  • adding/subtracting and multiplying/dividing fractions
• You can draw the above examples from the packet.
• Have students work on problems 3 and 4.

Roots [10 min.]

• Remind students what it means to be a square and to take a square root. Use this to build to what it means for a number to be a cube and to take a cube root.
• Give an example or two of cube roots (one positive, one negative is a good mix).
• Have students work on the remainder of the packet.

Comments on the Handout:

• Encourage them to not use calculators on this packet.
• Suggested problems to have them work through in groups on the board, if time allows: 7 - 12.
• For Problem 4, the work we expect to see is some factoring of the numerator and denominator and crossing out of common factors.
• Many students can be confused by Problem 5. Point out that the question refers to \textbf{least} common denominator, vs. just common denominator. Highly encourage groups to talk this one out.
• Problem 7 can be tricky for students; talk through the answer to this one.
• Problem 13 is also tricky, as a hint ask student to think about how much bread each person contributes, AFTER they've eaten their share. This problem is meant to be fun, and challenging; it is ok if students do not get all the way to this problem.

Suggestions:

In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.