11-5 6th grade math Counting Methods

Objective To use a tree diagram or the counting principle to find the total number of outcomes for an event. Why? To know how find a different way of finding possible outcomes CHOICES. You can use what you know from older lessons: Make a List, Make a Table, Draw a Diagram to help you make tree diagrams or the counting principle.

California State Standards SDP 3.1 : Represent all possible outcomes of compound events in an organized way (e.g., … grids, tree diagram) and express the theoretical probability of each outcome. SDP 3.3 : Represent probabilities as ratios, … and percentages between 0 and 100 …

Vocabulary Tree diagram Counting principle Counting grid A diagram used to organize outcomes of an experiment to make them easier to count. It sometimes does not matter which choice is addressed first, so more than one tree diagram can be correct. Count last row for outcomes. Different outfits can be made w/black, navy, or green pants with a white or yellow shirt. Counting principle If one choice can be made in m ways and a second choice can be made in n ways, then the two choices can be made together by m x n ways. m = pants; n = shirts m x n = pants x shirts = 3 x 2 = 6 Counting grid Use a grid to represent the different ways of sorting out the choices. outfits Black White Yellow Navy Green Pants Shirts Black (B) Navy (N) Green (G) Yellow (Y) BY NY GY White (W) BW NW GW

How to Use the Tree Diagram Make sandwich: Bread- pita, tortilla Filling- chicken, beef, veg. 6 ways to make a sandwich 1) Read the problem 2) Decide what choice will be first 3) Branch the second choices off the first. 4) Check your work sandwiches Pita chicken Beef Veg. tortilla Chicken

How to Use the Counting Principle 1) Read the problem 2) Decide what choice will be first to be m 3) The second choice will be n. 4) Multiply m x n 5) The product is the number of outcomes possible. Make sandwich: Bread- pita, tortilla Filling- chicken, beef, veg. Bread = m Filling = n Bread x filling = 2 x 3 = 6 6 outcomes of making sandwiches

How to Use the Counting Grid 1)Read the problem 2) Decide what choice will be first to be top row of the grid. Separate each choice into a column 3) The other choice will be separated into a row 4) ‘Multiply’ or put together the column and row of choices 5) The number of squares in the grid is the number of outcomes possible. Make sandwich: Bread- pita, tortilla Filling- chicken, beef, veg. 6 squares = 6 outcomes for making sandwiches Bread Filling Pita Tortilla Chicken PC TC Beef PB TB Veg. PV TV

How to Write the Probability of a Ratio 1)Read the problem 2) Decide the numerator as the number of favorable outcomes (asking). 3) Write the denominator as the total number of possible outcomes. 4) The number of squares in the grid is the number of outcomes possible. P(beef) Bread- pita, tortilla Filling- chicken, beef, veg. Bread x filling = 2 x 3 = 6 6 outcomes of making sandwiches Sandwiches w/beef = Bread x beef = 2 x 1 = 2 Ratio = 2 = 1 6 3 1/3 = .33 = 33%

Try It! 6 different ways to make a dish of noodles choices spaghetti Alfredo Veg. Meat fettuccine Draw a tree diagram from the chart below to show choices of sauce and pasta. Sauce Pasta Alfredo Spaghetti Vegetable Fettuccine Meat

2) P (spaghetti) = 1 P (veg. ) = 1 1 x 1 = 1 Total outcomes = 6 1 ≈ 0 2) P (spaghetti) = 1 P (veg.) = 1 1 x 1 = 1 Total outcomes = 6 1 ≈ 0.167 ≈ 16.7% or ≈ 17% 6 3) P (fettuccini) = 3 3 = 1 = 0.5 = 50% 6 2 Try Some More! 2) P (spaghetti with veg. sauce) 3) P (fettuccini)

Try a bit more… 4) 5 x 7 = 35 outcomes 5) 12 x 4 x 3 = 144 outcomes Use the counting principle to find the number of possible outcomes, taking one from each category. 4) 5 vegetables, 7 fruits 5) 12 colors, 4 posters, 3 sizes 4) 5 x 7 = 35 outcomes 5) 12 x 4 x 3 = 144 outcomes

Objective Review To use a tree diagram or the counting principle to find the total number of outcomes for an event. Why? You now know how find a different way of finding possible outcomes CHOICES. You can use what you learned from older lessons: Make a List, Make a Table, Draw a Diagram to help you make tree diagrams or the counting principle. If there are m possible outcomes for the first event and n outcomes for the second event, then there are m x n total possible outcomes.

Independent Practice Complete problems 6-15 Copy original problem first. Show all work! If time, complete Mixed Review: 16-20 If still more time, work on Accelerated Math.