PRACTICE TASK: How Much Mountain Dew?

APPROXIMATE TIME: ONE CLASS SESSIONIn this lesson, students will model multiplication of a fraction by a fraction using models and strategies such as partial products.

STANDARDS FOR MATHEMATICAL CONTENT

5.NF.4 Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)

b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

STANDARDS FOR MATHEMATICAL PRACTICE

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision.7. Look for and make use of structure.8. Look for and express regularity in repeated reasoning.

BACKGROUND KNOWLEDGE

Students need multiple experiences modeling fraction computation with manipulatives, but they also need to represent those models and explain the thinking that goes along with the manipulatives. Only then will students truly begin to understand the abstract ideas of algorithms associated with computation involving fractions.

The manipulatives and the resulting models students choose to use are often influenced by the context surrounding the problems they are given to solve. For this reason, there are very few instances when “naked” computation is a necessity for student understanding and learning of mathematics.

COMMON MISCONCEPTIONS

A common misconception is that students should learn the procedure of multiplying straight across the top and bottom (multiply the numerators, then multiply the denominators to find your product) before understanding what happens to fractions as they are multiplied together. By doing this, students are being robbed of a chance to make sense of the solutions they find. If students follow the procedure, then get the incorrect answer, do they know it doesn’t look right? Do they even care? By allowing students to model, we give students the opportunity to make sense of the mathematics embedded within the tasks. This will happen with the use of manipulatives, modeling the use of manipulatives through diagrams, and finally explaining what is happening with the models and writing abstract equations to match them. This model of learning mathematics is called concrete, representational, abstract or CRA.

ESSENTIAL QUESTIONS

● Which strategies do we have that can help us understand how to multiply a fraction by a fraction.

● How can understanding partial products (using the distributive property) help us multiply fractions by mixed numbers?

MATERIALS

● Video of How Much Mountain Dew? Task● 3-Act data for How Much Mountain Dew?

GROUPING

Individual/Partner Task

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION (All SMP’s apply!)

In this task, students will watch the video, then tell what they noticed. They will then be asked to discuss what they wonder or are curious about. These questions will be recorded on a class chart or on the board. Students will then use mathematics to answer their own questions. Students will be given information to solve the problem based on need. When they realize they don’t have the information they need, and ask for it, it will be given to them.

Comments

Anticipated questions students may ask and wish to answer:How many cups of Mountain Dew were consumed?How many cups of Mountain Dew are left?How much of the can of soda was consumed?How much of the can of soda is left?

Task Directions

Act I

Students are shown the video How Much Mountain Dew. Students are asked what they noticed in the video. The teacher records this information.Students are asked to discuss what they are wonder (or are curious about) as related to what they saw in the video. The teacher records these questions.

For the question “How many cups of Mountain Dew were consumed?”, students write down an estimate in their journals, then write down two more estimates – one that is too low and one that is too high. Next, students discuss the questions and determine the information they need. If they need some information, they ask for it. The teacher gives information that students need (see Act II, below).

Act II

2/3 of the glass of Mountain Dew was consumed 1 can of Mountain Dew contains 12 fluid ounces 8 fluid ounces = 1 cup

Once students have what they need, they can begin to solve the problem.

Act IIIStudents share their solutions and strategies.

Compare and share solutions and strategies. How reasonable was your estimate?

What might you do differently next time?

CommentsStudents need the opportunity to work with manipulatives on their own or with a partner in order to develop the understanding of what happens when fractions are multiplied by fractions. From the manipulatives, students will be able to move to pictorial representations of the model, then more abstract representations (such as sketches), and finally to abstract representation of multiplication using equations. It is important to remember that this progression begins with concrete representations using manipulatives.

FORMATIVE ASSESSMENT QUESTIONS

● What models did you create? ● What organizational strategies did you use? ● What product/area does your model (do your models) represent?

DIFFERENTIATION

Extension● Have students use different models to solve the same problems and share these

models with the class. ● Give students another situation involving fractions and mixed numbers to see what

models they use to solve the problem.

Intervention● If necessary, allow students to use a multiplication chart or other cueing device if full

mastery of the basic multiplication facts has not yet been attained.