## WHAT'S FOR LUNCH? A COMBINATION SENSATION! Grades K-2

The purpose of this lesson is to allow young students to explore the concept of combinatorics, the science of counting and finding possible solutions, to describe patterns of arrangements. The activity is designed to allow students to use information they learned in previous lessons about the food groups and healthy food choices. The activity plan assumes that the students have already studied about food groups and are able to identify some foods in each of the groups. Although it would be helpful to maintain the continuity of the lesson by teaching it all in one day, this lesson may require more than an hour to complete. If dividing the lesson into two days, conduct Pre-Viewing activities on one day and Viewing and Post-Viewing activities the second day.
Math Talk #109: Let Me Count the Ways: Counting With Combinatorics
Students will be able to:
• predict the number of possible combinations of food choices
• work together as a group to determine a reasonable solution for the number of possible combinations.
• identify the food groups for a menu of food items.
• identify at least two different combinations of the food items by following guidelines on food choices.

• Demonstrate an understanding of probability and statistics.
• Estimate solutions to a problem situation.
• Determine solution strategies and analyze or solve problems. Science Objectives:
• Interpret scientific data and/or information.
• Make inferences, form generalized statements and/or make predictions using scientific data and/or information.
Per class:
• poster or overhead transparency of Food Guide Pyramid and food items
• yellow, red and green construction paper, 5 sheets each
• black, blue and brown construction paper strips, 5 sheets each

Per student:
• copy of the tray
• copy of food items
• scissors
• glue
• sticky note
• choice
• cereal
• milk
• food
• pasta
• fats
• food pyramid
• serving
• sweets
• food group
• fruit
• combination
• variety
• vegetable tray
• grain
• protein
• rice
• meat
• dairy
9 12 14
8 9 10 12 13 14 15
6 8 9 10 11 12 13 14 15
Tell the students that they will be viewing a short story that is about making combinations and looking for all of the possible combinations. Ask the students to listen and watch for ways Super Guy and the sales person organize their information to find all of the possible combinations or ways of putting things together.

yellow capes green capes red capes

black belts

brown belts

blue belts

Tell the students they can sort their food choices on their trays in a similar way. Ask one student to identify food on his or her tray and use this as a way to begin sorting. This could be something like all the trays with apples and all the trays without apples. Have students place their trays in one of these two groups on the floor or attached to a display board. The trays will be in a formation something like this. Trays with apples Trays without apples

Now ask another student to identify a different food on the trays, such as chicken. Divide each of these two groups of trays into two groups. Trays with Trays without apples

Trays with chicken

Trays without chicken (fish)

Have the students examine the trays to determine if any of the trays have all the same food choices. If there are any trays with exactly the same food choices, these can be stacked on top of each other. Have the students count to find out how many different sets of food choices they see. Ask the students if they think they have found all the different ways to choose food selections for lunch. If they think they have, use the display to show them a combination they did not choose. Have the students focus their attention on the trays which have chicken. One group has trays with apples. What foods could go with apples? Have the students sort the trays in a row. Record these combinations either with food pictures or with letters to represent the foods (if this will be understood by the students). Your combinations may look like this. (c=chicken; a=apple; o=orange; b=broccoli; l=lettuce salad)
C A B C A O C A L
Are there any other foods that could go with chicken and apples? (no) What other food could you choose to go with chicken? (Students name one: broccoli, for example.) Have students look at the trays to find possible combinations with this food.
C B A C B O C B L
Is there one of these combinations that is the same as another we listed? (Yes. CAB is the same as CBA.) Erase or put an X on that combination. Are there any other foods that could go with chicken and broccoli? (no) What other food could you choose to go with chicken? (Students name one: oranges, for example.) Have students look at the trays to find possible combinations with this food.
C O A C O B C O L
Are there combinations that are the same as another we listed? (Yes. COA is the same as CAO. COB is the same as CBO.) Erase or put an X on those combinations. The list of combinations should look something like this.
C A B C A O C A L
C B O C B L
C O L
How many different combinations are there when the meat/protein selection is chicken? (six) What could you choose for meat/protein if you did not choose chicken? (fish) If there are six choices with chicken, how many do you think there will be if you select fish? Have the students examine the trays with fish and see if they can sort them to see if they are right. You may wish to show the students how they can change the list of combinations with chicken above by changing the C to F.
F A B F A O F A L
F B O F B L
F O L
If there are six combinations of food selections with chicken and six combinations of food selections with fish, how many different possible combinations of food are there? (twelve) Do you have to know how to make them all? (no) Review the main points of this lesson. When do we have food choices to make? (in the cafeteria, at restaurants, at home, when shopping) What kinds of foods should we choose if we want to stay healthy? (foods from all food groups on the food pyramid) How do we know which food to choose? (foods from food pyramid, foods we like) Is there more than one way to choose foods in the cafeteria? (yes) How might we find out about different combinations? (try it out, use pictures, write letters for the foods, list the combinations) How would we know when we have all the combinations? (organize the information) When else might we want to know about combinations? (making teams for games, choosing clothes to wear, making things with different colors)
Have the students examine the local cafeteria menu. Does it include healthy foods from all groups on the food pyramid? Does it limit fats, salt and sweets? Students can write a letter to the cafeteria manager or menu planner to suggest ways to improve the lunch menu. Use telecommunications to communicate with classes in other schools in different parts of the state or the country. Ask them to share how their school cafeteria allows students to make food selections. Ask them to share a recent menu. How does this menu compare with the one at our school? Do students have choices which allow them to choose from all food groups on the food pyramid?
Writing: Have each student write a sentence to tell about his or her food choices. The sentence may read something like: "On my tray, I choose chicken, lettuce salad, an apple, a roll and milk." Have each student read the sentence to the class. For younger students, they may use their paper tray of food choices to talk about the food choices.
Science: You are planting a school garden. You have learned that there are two kinds of trees and four kinds of flowers which are especially easy to grow in your area. If you wish to plant one tree and two different kinds of flowers in your garden, how many possible combinations would there be? [Note: After doing this activity, some students may recognize this as the same problem posed in the lesson with the food choices. There are 12 combinations.]
Health: The students will be provided the materials to make a trail mix. There are five ingredients available and each student may select three of these ingredients for the trail mix. How many different possible combinations are there? (10)
ABC ABD ABE BCD BCE CDE
ACD ACE BDE
Health: Have each student prepare one day's food selections for the cafeteria menu. When the food selections are prepared, have other students check to see that a variety of foods from the food pyramid are available. Submit the class menu to the cafeteria manager or menu planner.
Mathematics: Have the students investigate another instance of combinatorics. For example, if Justine has four shirts (yellow, white, red, and green) and three pants (black, brown, and blue), how many different combinations of shirt and pants can she make? Have students use linking cubes to show the possible combinations.
Social Studies: Have students use a map to locate five different cities or sites which are within a relatively short distance of their home that they would like to visit. List these places on the blackboard. If they were able to visit two of these places on a vacation, which would they choose? How many different possible combinations of two places would there be? Could they visit both places they chose in one day?

NOTE TO TEACHER

For the esl learner: The student who is learning English as a Second Language will have the opportunity to use many common words in this lesson. Many of the vocabulary words are included with the pictures of the food items the students will use to complete the activity. As you are reviewing the food groups in the food pyramid, you may wish to explain or have students identify examples of foods in each group to help clarify these words. Actual food items for students to sort or pictures of food items for students to place on a blank food pyramid could also help students relate the vocabulary to real-life items. Try to include food items which are familiar to your students. These could include foods served in their homes, such as rice or tortillas, as well as foods they are familiar with from school, such as local fruits and vegetables. Provide an opportunity for students to write sentences about and read their combinations of food choices to others in the class to support written and oral communication in English.

Background information on dietary guidelines: The USDA's Food Guide Pyramid provides the following information about healthy food choices. Americans above the age of two should eat a variety of foods, maintain a healthy weight, choose a diet low in fat and cholesterol, choose a diet with plenty of vegetables, fruits, and grain products, and use sugars and sodium in moderation. The food pyramid emphasizes the five food groups located in the three lower levels of the food pyramid. Each of these foods provides some of the nutrients you need daily. Since no one food is more important than another, a variety from each of the groups is needed for good health. The base of the pyramid includes foods derived from grain, such as bread, cereal, rice, and pasta. The most servings come from this group. The next level includes foods which come from plants: fruits and vegetables. The third level includes foods which, for the most part, come from animals and are sources of protein and calcium. Meat, poultry, fish, eggs, beans, and nuts are in one group on this level. Dairy foods, such as milk, yogurt, and cheese, are also on this level. The foods at the tip of the pyramid, which include fats, oils, and sweets, are recommended to be used sparingly.

Background information on combinatorics: "Combinatorics" is the mathematics of counting. More generally, it is concerned with problems that involve a finite number of possibilities and it attempts to answer one or more of the following questions: - Does a solution exist? - How many solutions are there? - Is there an optimum solution?". (E.W. Hart, 1991, p.70) In this lesson, each student will find at least two combinations of food items. (Does a solution exist?) After examining the combinations of others in the class, each student will predict the total number of possible solutions. (How many solutions are there?) Each student will paste one combination on a paper tray to show a choice. (Is there an optimum solution?) A total number of possible combinations may be identified.

### Master Teacher: Wayne Gable

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