Estimation

Classroom Activities

Each lesson of the unit will take two to four class sections of 50 minutes to complete. If all the lessons are done together, this estimation unit will probably spread over about four weeks to five weeks. There are many essential 6 ^th grade math topics involved, so I will likely spread out the lessons as we come across them through out the semester. The cumulative project may be assigned closer to the end of the school year as we prepare for the zong cooking lesson and celebrate the Dragon Boat Festival. I hope the students will remember the things they have learned about estimation when they enjoy the zong.

Lesson 1- Population

Exploration/ Motivation

Play a game of hangman using the U.S. population figure 303,824,646. Students have to use the correct place value names to make a guess of the digits. After they have guessed the number, have a short discussion about how accurate this number is. What are the factors that may change this figure? We want the students to understand that this figure is only an approximation. It is nearly impossible to know the number of people in the U.S. precisely at any given moment due to births, deaths, migrations, etc. An option is to incorporating a quick computer search lesson by looking up the information online with the students in class.

Direct Teaching

Rewrite 304,583,860 in the expanded form and in the order of magnitude notation. Explain the notation and ask the students to describe the pattern they see in the notations. Compare the values of the two 3s and 6s in the number. Lead students to compare the places by thinking in multiplication terms.

Now introduce the rounded figure of the U.S. population (3 x 10^8). Go over the parts of the notation and their meaning. Ask students if it makes a big difference if we take away each one of the places starting from the largest place. The effect is huge with the hundred million place. Eliminating the ones place does not matter much. Furthermore, Show students how to write 3 x 10^8in different magnitudes.

Guided Practice

Show the population of Hong Kong (7 x 10^6) and Taiwan (2 x 10^7). First ask them to decide which place has more people just by looking at the numbers. Next, have the students examine the numbers more closely. How are the coefficients different? How are the orders of magnitude different? How do the numbers compare to each other? Again, encourage them to think in multiplication, not in subtraction. Rewrite the numbers to the same order of magnitude with the students and show them how to divide by subtracting the exponents.

Reinforce the understanding of the powers of ten by giving data for other places and have students compare the numbers. Provide pairs of numbers with either the same coefficients and/or the same order of magnitude. Also, compare other pairs of populations with a larger multiple. The difference may be nearly the same as the larger country. The smaller country might even be lost in the rounding error of the larger country if we round to a large place. The 20 million population figure for Taiwan is rounded from 23 million. The amount lost in rounding is nearly half the population of Hong Kong.

Independent Practice/ Assessment

As a conclusion to this lesson, give students a list of population figures and country names. Have the students work cooperatively to put the population numbers in sequence from smallest to largest and match the country names with each population. Encourage them to think about what knowledge do they have that would help them to make the match ups. A list of countries and population figures is included in the appendix. An extension activity can be to have students find the top ten countries of rice consumption and or production.

Lesson 2- Ratios and Proportions

Exploration/ Motivation

Working in small groups, let students dissect a zong to discover the ingredients in it. Each group must discuss and device a plan of action and have it approved from the teacher before starting to dissect the zong. The plan should contain their steps in sequence and each person's responsibility. If it is possible, find zong in different sizes to show the variations. That will provide the opportunity to discuss about finding an average and what it means to estimate. Each group will make a poster and give a report at the end of the activity that must include these information: the net weight of the rice, the weight of the other ingredients, the total weight of the zong and whether this figure includes the bamboo leave wrapping and string, and the percentage of the rice compared to the total weight. The class can compare their findings to recipes. At the end, the class as a whole will determine an average amount of rice to go into one zong.

Direct Teaching

After the class finds the weight of the rice needed to make one zong, we will learn how to to do estimations by using proportional reasoning. The problem solving question is how much rice to prepare for our zong cooking lesson. Ask the students to think about what information do we already know and what other information do we need. We will know the amount of rice for one zong from the dissection activity. We need to determine how many zong to prepare or how many people to feed. Show students how to set up proportions correctly and how to solve it using cross multiplication. Some students may figure out that we can simply multiple to get the quantity of rice. However, some students may be confused about when to multiply or divide to solve these problems. Setting up the proportions gives us a way to trace out steps and check our thinking.

Guided Practice

Practice the proportion skills using the U.S. population. The new problem solving question is how much rice do we need to make one zong for each person in the U.S.? This time, let students come up with the information that they already have and the information that is missing. Guide them through the steps to set up the proportion. Ask students to explain how it works. Talk about the possible mistakes in the proportions and how to check if it is correct.

Independent Practice/ Assessment

Students can choose a country from the list of populations from the first lesson to find the rice quantity needed. An extension may be to include the frequency element. Instead for preparing one zong for each person, how does the estimates change if two people would share one zong or eating zong for three meals a day?

Lesson 3- Area and Volume Investigation

Exploration/ Motivation

Have students use Cuisenaire rods to build squares for various areas. Fill the square with one layer of rice and measure the weight of the rice. To help make sure the rice is layered one grain thick, I suggest using a ruler or paper to press down. Repeat this with squares of several sizes. Ask students to record their measurements and make observations of the data. Discuss how they can predict the difference in the mass of rice to fill the different squares. Again, lead students to think about the change in multiplication terms. After investigating with area, perhaps in another session, students can make cubes out of cardboard and centimeter paper to explore about volume of different sizes.

Direct Teaching

The problem solving question for this lesson is what is the volume requirement of the storage place to hold the rice for our zong cooking lesson? We already know that we need 5 kg of rice for our cooking lesson. We need to know the volume to weight ratio to complete a proportion for our estimation. We can either use information from research or we can use figures from our volume investigation to write the proportion to solve this problem. This is also the good time to show students how to decide on the appropriate units and how to do unit conversions.

Guided Practice

Again, here the problem solving question relates to the U.S. population. The question is what is the volume of a storage shed to hold 15,000 metric tons of rice? Since the number is so huge, we may want to help students break down the problem to more manageable parts. We can first find out how big is the volume for 1 metric ton of rice. Then, we can use the new ratio to calculate for the 15,000 metric tons.

Independent Practice/ Assessment

Here, students can use the volumes of rice they estimated earlier in lesson 2 to practice finding volume of the storage spaces. A variation of this exercise is to put the rice into boxes. Determine the volume of the box and how many boxes of rice we have. Use the volume of the boxes to determine the total volume. Or convert the usual 50lb bags of rice to metric measures and determine how many bags of rice we would have. Then calculate how many bags will fit in our classroom and how may classrooms we need to store all the bags. We can also find the volume of a storage shed in our school and calculate how many sheds we need and how big is the land to fit all the sheds.

Cumulative Rice Project

Students will work in small groups to complete a project on estimations relating to rice.

They will create their own problem solving question, define the smaller questions, devise a plan, solve the problem, organize the information and present it in a report to the class. Each group will customize a rice problem by selecting a place in the world, choosing a recipe to use and deciding on a time criterion for the estimation. They must estimate for the rice quantity needed in weight for the population and devise a storage solution for the rice.

Alternatives for basic students may include allowing them to use the rate of the rice usage in the zong activities, choosing a more manageable size of population to work with and/ or estimating for a more manageable time frame. Options for proficient and advanced students may include researching other elements relating to rice, such as water consumption, comparing rice yield and land usage of different states or countries. For students who are highly interested, they may expand to other grains or other types of foods.