Tennessee Department of Education Task: Papa’s Pizza 4th Grade You are the owner of Papa’s Pizza, a small pizzeria in Tennessee that sells pizza by the slice. Each pizza has 8 slices. A customer comes into your store at the end of the day and wishes to purchase 1 pizza. You send an employee to the kitchen to get a pizza, and she tells you that there are 4 slices of pizza and ½ of another pizza left. (1) Draw a model for each pizza to show how much pizza is left in each. (2) Write the portions of pizza as fractions. (3) Using the models, how do the 4 slices and ½ pizza compare? Record the results of your comparison using the symbols >, <, or =. (4) Determine whether you have enough pizza to sell the customer a whole pizza by showing how much pizza you have in a diagram and tell how much pizza you have to sell. Extension: What if your pizzas were sliced into 16 pieces? Complete numbers (1) through (4) using pizzas divided into 16 slices. Teacher Notes: A goal of this task is to help students understand that 4/8 is equivalent to ½ by using models. For Question 3, if students simply say that the two amounts are equal by observing their models, encourage them to prove that by finding common denominators and/or simplifying 4/8 to ½. Common Core State Standards for Mathematical Content Common Core State Standards for Mathematical Practice 4.NF.1 Explain why a fraction a/b is equivalent to a fraction 1. Makes sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. (n  a) / (n  b) by using visual fraction models, with attention to 4. Model with mathematics. how the number and size of the parts differ even though the two 6. Attend to precision. fractions themselves are the same size. Use this principle to 7. Look for and make use of structure. recognize and generate equivalent fractions. 4.NF.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the result of comparisons with symbols >, =, or <. 4.NF.3a Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 1 Essential Understandings  The rational numbers allow us to solve problems that are not possible to solve with just whole numbers.  One interpretation of a rational number is as a part‐whole relationship.  Any rational number can be expressed as a fraction in an infinite number of ways.  The interpretations of the operations on rational numbers are essentially the same as those on whole numbers, but some interpretations require adaptation, and the algorithms are different. Explore Phase Possible Solution Paths Assessing and Advancing Questions (1) This model could be drawn as a pizza (or rectangle) with 8 Assessing Questions pieces, 4 of them shaded, and another divided in half with one  Tell me how you know that your models represent the portions of half shaded. pizza that you were asked to model. (2) The student may write the fractions 4/8 and 1/2 to represent  How do you know that 4 pieces of pizza and 1/2 of a pizza are the the different portions of pizza. same amount? (3) The student should determine that 4/8 and 1/2 represent the Advancing Questions same fraction of pizza, and, hopefully, that  Can you think of a different fraction you could have used to represent (1 4) / (2  4)  4 / 8 . the two fractions you used in step (2)? (4) The student should notice that 4/8 of a pizza added to 1/2 of a  How are 4/8 and 1/2 related? What could you do to 1/2 to make it pizza of the same size is a whole pizza, so the customer can into 4/8? purchase a whole pizza. Assessing Questions Alternate Solution  Tell me how you know that your models represent the portions of (1) The model could be drawn as a pizza (or rectangle) with 8 pizza that you were asked to model. pieces, 4 of them shaded, and another divided into 8 pieces  How do you know that 4 pieces of pizza and 4/8 (or 1/2) of a pizza are with half of them shaded. the same amount of pizza? (2) The student may write the fractions 4/8 and 4/8 OR 1/2 and Advancing Questions 1/2 to represent the different portions of pizza.  Can you think of a different fraction you could have used to represent (3) The student should determine that 4/8 and 4/8 (or 1/2 and the two fractions you used in step (2)? 1/2) are the same fractions.  How are 4/8 and 1/2 related? What could you do to 1/2 to make it (4) Thus, there is a whole pizza available to sell to the customer. into 4/8? Possible Student Misconceptions  That the fractions refer to different wholes. Models could lead  Are all of your pizzas the same size? Are the wholes you are drawing them to this assumption if they are not drawing the same size the same size? pizzas (or rectangles).  What is your whole in this problem? A slice or a pizza?  That the 4 slices are 4 wholes, rather than parts of a whole 2 pizza. Entry/Extensions If students can’t get started… If students finish early… Assessing and Advancing Questions  How could we represent slices of pizza in a drawing?  How many parts are in your whole?  How many pieces of the whole do you have? Assessing Questions  Explain to me, in writing, how you know that each of your steps is correct. Advancing Questions  What if your pizzeria divided their pizzas into 16 pieces, instead of 8?  Can you do steps (1) through (4) for this situation? Discuss/Analyze Whole Group Questions The Fraction Model (Step 1)  Someone show us how you modeled the pizza in Step (1).  Did anyone do it differently?  Are both of these correct? The Fractions (Step 2)  What fractions did you write in Step (2)?  Did anyone write different fractions?  Are both of these correct? Comparing the Fractions (Step 3)  What is the relationship between the fractions?  Does anyone agree/disagree? Answering the Question (Step 4)  Who thinks that we have a whole pizza to sell to the customer?  Who thinks we don’t?  Why? Equivalent Fractions  Discuss the concept (n  a) / (n  b) =a/b, using the models and fractions written by students in this problem. 3