Preparing the Investigation
Students should be familiar with using both nonstandard and standard units of measure and have been introduced to the concepts of area and volume.
Before introducing this baking activity, the teacher should:
- know how the cost of required ingredients will be covered;
- know the package directions for making the brownies, including additional ingredients required and their amounts, pan sizes, serving sizes, and number produced;
- have access to an oven or microwave in which to bake the brownies;
- have a list of all ingredients, utensils, and equipment and the quantities of each item needed for preparing and baking brownies. Utensils and equipment should not be shared by groups working simultaneously. Measuring cups for the oil and water should be made of glass or clear plastic and be designed for use with liquids. Teacher directions and student questions follow.
Structuring the Investigation
Each small working group of students should have two different-sized cake pans (8" × 8" × 2", or 9" × 9" × 2", or 9" × 13" × 2"), and a bowl or a pitcher containing 2 1/2 to 3 cups of water. The water will represent the batter. Ask students to consider pouring a given amount of water from the pitcher into each cake pan. Have them first make a guess and then actually do the pouring. In doing so, students can answer questions such as:
- Which pan will provide the deeper measure of water?
- Which pan will provide the more shallow measure of water?
- Which pan will provide the greater top surface?
- Why? How can you tell?
Put each pan on a piece of paper, draw the outline of the pan bottom, and then cut out the shapes. Make several sets of those shapes. Ask students questions such as: Which pan requires the larger piece of paper? Which pan requires the larger number of square colored tiles to cover the bottom? Why? Without employing any measuring instruments, fold or cut them into two equal pieces. Try dividing a piece into three equal pieces, six equal pieces, and twelve equal pieces. In doing so, students can answer questions such as:
- Which cuts or folds are the most accurate?
- Can you get reasonably accurate pieces more easily for some numbers than for others? Why?
- Which division can be completed most accurately?
- Which divisions are only approximations of equal pieces? Explain?
Pose the following scenario to students:
Would it be easier to complete the previous activity if the original shapes had been drawn on rectangular-grid or graph paper? Why? Use grid paper and repeat the previous activity. Is it any easier? Why? Compare your sets of pieces with corresponding ones from another group. What do you notice? How easy will it be to cut one of the rectangles into exactly seven equal pieces? Why?
Note: So far, the word equal has been used; however, students may be thinking congruent. A 1-inch-by-4-inch rectangular shape is equal in area but not congruent to a 2-inch-by-2-inch square shape. All congruent shapes have equal areas, so those shapes are also equal. However, all shapes with equal areas are not necessarily congruent.
Ask students, "Without measuring, is it easier to fold or cut a rectangular shape into two equal pieces or three equal pieces? Why?"
Pose the following second scenario to students:
If you have a rectangular piece of paper and no measuring tool, which of the following could probably be done the fastest:
- mark the paper into six equal pieces,
- mark it into seven equal pieces, or
- mark it into eight equal pieces?
If you choose to demonstrate or allow students to practice on their own, you may mark the paper with a crease if necessary. Students should give reasons to support their choices. Ask students which of the three markings listed above (a, b, or c) would probably be the most accurate, and why.
Once students have discussed these measurement concepts, it is time for them to bake (and eat!) some brownies. This may be done during a follow-up class session if there is not enough time today.
Show students how to break an egg and to check it for freshness before adding it to the mix.
Ask each group to read the package directions and discuss among themselves what is to be done and the proper order in which to do all steps.
Each group decides which member will be responsible for individual steps in the preparation. Ask the group to complete a sign-up sheet. All members are to help with cleanup. Job assignments need to be recorded and posted at each workstation.
Sign-Up Sheet for Baking Duties
Brumfield, Emalou and John Firkins. "Ideas." Arithmetic Teacher 41 (February 1994): 309-316.
Questions for Students
Refer to the Instructional Plan.
- How did you determine whether students were engaged with this activity?
- What manipulatives could you use to help students who were struggling?
- How could you modify this lesson to incorporate more measurement techniques, problem solving skills, and the concept of equality vs. congruency?
- Use problem solving skills, measuring techniques, and food preparation experiences to practice various math concepts.
- Prepare, after determining minimum amounts of ingredients required, a
commercial brownie mix and serve equal portions to all class members.
NCTM Standards and Expectations
- Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
- Understand such attributes as length, area, weight, volume, and size of angle and select the appropriate attribute.
- Understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems.
Common Core State Standards – Mathematics
Grade 3, Algebraic Thinking
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8.
Grade 3, Algebraic Thinking
Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers.
Grade 5, Num & Ops Fractions
Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Grade 5, Num & Ops Fractions
Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
Grade 4, Algebraic Thinking
Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
Common Core State Standards – Practice
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.