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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.
ITV Series
Math Talk #109: Let Me Count the Ways: Counting With Combinatorics
Learning Objectives
Students will be able to:
Per class:

Per student:
Pre-Viewing Activities
[Note: A major portion of this lesson will involve activities done prior to the viewing of the ITV program. The segment from Math Talk which is used in the Viewing Activities section of the lesson deals with combinatorics and will help students organize their information and make predictions about and verify the number of possible combinations of food items.] Remind the students that they have been learning about healthy eating choices and that people should eat a variety of good foods in sensible portions. Bring to their attention a blank display of the food pyramid, which is something the class has examined in an earlier lesson. Ask the students to identify the food groups. As each group is identified, place a label or write the name on the food pyramid. Show the students a picture of a common food item from the school's cafeteria menu. Real food items might be used with younger students. Have the students name the food item and identify the food group to which it belongs. Place the picture of the food item on the blank food pyramid. Continue until each of the food groups has several examples in it. You may also wish to include some food items that are composed of food from different food groups, such as pizza (bread, vegetables, dairy) or enchiladas (bread, vegetables, meat, dairy). Choose the menu for one lunch from the school cafeteria menu. You may wish to choose a recent lunch or one to be served in the near future. Determine the food groups to which each food item belongs. [Note: Many school cafeterias are allowing children to make selections among a small variety of food choices. The following activity is based on this idea. The students will select one meat/protein dish from two choices and two fruit or vegetables from a selection of four fruit or vegetable choices. Each lunch will also include a roll (one bread is served, no choice) and milk (dairy, no choice). The activity could be adapted to fit your school's cafeteria process. If this is not appropriate, you may wish to place the activity within the context of making choices at a local restaurant or cafeteria.] Show the students the display of the cafeteria tray. Ask the students to identify where each of the food items from the cafeteria menu would fit on the tray, in both space and food group. You may wish to list the food items in a manner such as this. meat/protein (1 choice): chicken, fish fruit/vegetable (2 choices): broccoli, salad, apple, banana bread: roll dairy: milk Ask the students to use the paper cafeteria tray and the pictures of the food choices to show one choice of selections for lunch. The students will place one food item in each space on the cafeteria tray. Students must choose one meat/protein selection and two fruit or vegetable selections. They may choose either two different fruits, two different vegetables, or one fruit and one vegetable. They cannot choose two of the same foods (for example, two servings of oranges). Ask the students to tell other students around them what choices they made. Did everyone make the same choices? Why did this happen? Have two students name the food selections they choose. Place these selections on the display trays. Ask students to identify the similarities and differences between the two meals. Discuss the concept of food preferences and how that will effect the choices people make. What other factors might change the choices you make? Have the students examine the food selections again and have them make a final choice for the lunches they would like to eat. These selections should be glued in the appropriate places on the food tray. [Note: With some groups of students, you may wish to conduct this choice and pasting activity in a small group to assist students in locating the foods correctly. With a large group, you may wish to have the students work item by item with you as you model. Your oral directions may be very specific to emphasize vocabulary or to encourage the students to practice following directions. You may say something like this: "Take a milk. Paste it in the space next to the fork. Do you want to eat chicken or fish? Pick up one and paste it in the space on the bottom next to the milk."] Have the students examine the trays of others near them. Ask them to talk with their neighbors about how their choices are the same and different. Ask the students to tell you something they noticed which is the same about every person's tray. The students should be able to recognize that every tray has milk and every tray has bread. Tell the students that since each tray has both milk and bread, those food items will not need to be considered when we begin sorting. Ask the students to think of a way their trays could be sorted. Have the students discuss how the trays could be sorted with the others near them. For example, they might think of putting all the trays with fish together. Ask the students to predict the number of possible ways the lunch tray could be filled with different food selections. Give each student a small sticky note. Have each student write the prediction on the note. Have the students place the sticky notes on the wall or blackboard to make a graph of the predictions. The graph may look something like this.
9 12 14
8 9 10 12 13 14 15
6 8 9 10 11 12 13 14 15
Focus Viewing
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.

Viewing Activities
Begin the Math Talk video when the black and white sign labeled "Water Works" is shown. Show the segment through the explanation of the problem at the Water Works and pause after Super Guy knocks on the occupied phone booth, turns away and says, "What am I going to do?" Ask, "What is Super Guy's problem?" (water is leaking from the wall, he wants to help stop the water, he can't find a place to change into his Super Guy suit) Tell students to watch the next segment to see where Super Guy goes to change into his Super Guy suit. Resume the video to continue through the segment where Super Guy goes into the clothes store to change into his suit. The sales person points out that Super Guy has a problem: he forgot his cape. Pause where Super Guy says, "I don't have time to get a new one. Destiny calls!" Ask, "Where did Super Guy go to change his suit?" (the clothes store) Ask, "What did the sales person tell Super Guy?" (He forgot his cape.) Tell the students to watch the next segment to see how Super Guy will fix his problem of having no cape. Resume the video through the segment where Super Guy notices the capes hanging on the rack. Pause after the sales person says, "Well, they're all very nice capes, sir." Ask, "What did Super Guy see?" (capes hanging on the rack) Ask, "What color capes did Super Guy like?" (yellow, red, green) [Note: If students do not recall the colors of capes, rewind through capes hanging on the rack and replay.] Display sheets of yellow, red, and green construction paper to represent the capes. Tell the students to watch the next segment to see what else Super Guy wants. Resume video through the segment where the sales person shows the black, brown, and blue belts and pause where Super Guy says, "Oh, I could wear the blue belt with the yellow cape." Ask, "What else does Super Guy want?" (a belt) Ask, "What colors are available in belts?" (black, brown and blue) [Note: If needed, rewind and replay this segment to remind students of the colors available in belts.] Display strips of black, brown, and blue construction paper to represent belts. Ask, "What combination does Super Guy think of first?" (yellow cape with blue belt) Display strip of blue construction paper on across yellow construction paper. Ask, "Do you think that's the only thing Super Guy could choose?" (no) Ask, "What other combination of cape and belt might Super Guy choose?" Allow students to name several different combinations and use the construction paper and paper strips to display these. Tell students to watch the next segment to see if Super Guy and the sales person can think of different combinations of capes and belts as the students did. Rewind briefly to where Super Guy is holding the green cape. Resume the video through the segment where they talk about the number of combinations and pause where the sales person says, "But we can figure it out." Ask, "How many combinations do you think there will be?" (various predications) Ask, "How do you think we could figure it out?" [Note: Lead students to recognize that they could use the construction paper pieces and strips to show the combinations.] Ask, "What combinations of one cape and one belt could we make?" As students name a combination, use construction paper and strips to display them. Their combinations may include duplicates or not include some possibilities. Be sure to display these as this is part of the learning to make combinations. Tell the students to watch the next segment to see how many combinations Super Guy and the sales person make. Resume the video through segment where the combinations are made on the display rack and pause where Super Guy says, "Now, have we forgotten anything?" Ask, "How many combinations did they make? (eight) Rewind the video to the view of Water Works sign and people against wall. Tell students to watch to see which combinations match the ones they made. [Note: If students have not recognized that some combinations are duplicated at this point, you may wish to lead them to see this before viewing the next video segment.] Resume and pause where the sales person says, "Well, let's see if we can get all of this in some sort of order." As you watch the segment this time, remove the construction paper cape and paper strip belt combinations from the display. Ask, "How many combinations did we name that Super Guy and the sales person also found?" Count those removed to determine how many were the same. Ask, "They had eight combinations. Did they have some combinations we didn't name? Did we have combinations they didn't name? How many?" Tell students to watch the next segment to see how Super Guy and the sales person organize the capes and belts and to see if they find all of the possible combinations. Resume the video through segment where they move the combinations and pause where the sales person says, "Wait a minute. What goes here?" Ask, "How did they organize the capes?" As the students tell how all the yellow, green, and red capes were put in different columns, put the construction paper capes in columns on the display. As the students tell how all the black, brown, and blue belts were put in different rows, put the combinations in rows on the display.
yellow capes green capes red capes

black belts

brown belts

blue belts

Ask, "Can you figure out what is missing?" (red cape with a black belt) Ask, "How can you figure this out?" Ask, "How many combinations did you find?" (nine) Ask, "Do we have all the combinations?" (yes) [Note: With younger children, it may not be as important that they find every possible combination as it is that they realize that different combinations are possible and that different choices can be made.] Tell students to watch the next segment to see if they are right about the missing combination and which combination Super Guy selects. Resume the video and continue through segment where Super Guy puts on the cape and belt and stop at the Adventures of Super Guy logo. Ask, "Were we right about the missing combination?" (yes, red cape with a black belt) Ask, "How many possible combinations were there?" (nine) Ask, "Which combination did Super Guy select?" (red cape with a black belt) Ask, "Why do you think he selected this combination?" (He liked it.) Ask, "Why do people select things like clothes or food?" (They like them. They taste good. They want them.)
Post-Viewing Activities
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)
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.
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.
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.
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.
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)
Action Plan
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)
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?


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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