Use area models to prove that , and What conclusion can you make about and Explain. Note: This Application Problem builds from Lesson 5, where students decomposed unit fractions using area models to show equivalence. Consider leading a discussion with a question, such as, “Why can you show = = on one model, = = on another, and = on another?” Or perhaps lead with a question, such as, “Why can’t you show = = on the same area model?” Concept Development (30 minutes) Materials: (S) Personal white board Problem 1: Use an area model to show that . T: Draw an area model representing 1, and then shade . T: Discuss with a partner how you can use this model to show the decomposition of 3 fourths into eighths. S: We could draw a line so that each of the fourths is split into 2 equal parts. That would give us eighths. Drawing a line will make each unit into 2 smaller units, which would be eighths. T: How many eighths are shaded? S: 6 eighths. T: Work with a partner to write an addition and a multiplication sentence to describe the decomposition. S: is equal to T: What do these addition and multiplication sentences tell you? S: The shaded area didn’t change. It’s still the same amount. The number of pieces increased, but the size of the pieces got smaller. Adding together all of the smaller units equals the total of the larger units shaded. Multiplying also equals the total of the larger units shaded and is easier to write out! Problem 2: Draw an area model to represent the equivalence of two fractions, and express the equivalence as the sum and product of unit fractions.
Use area models to prove that , and What conclusion can you make about and Explain. Note: This Application Problem builds from Lesson 5, where students decomposed unit fractions using area models to show equivalence. Consider leading a discussion with a question, such as, “Why can you show = = on one model, = = on another, and = on another?” Or perhaps lead with a question, such as, “Why can’t you show = = on the same area model?” Concept Development (30 minutes) Materials: (S) Personal white board Problem 1: Use an area model to show that . T: Draw an area model representing 1, and then shade . T: Discuss with a partner how you can use this model to show the decomposition of 3 fourths into eighths. S: We could draw a line so that each of the fourths is split into 2 equal parts. That would give us eighths. Drawing a line will make each unit into 2 smaller units, which would be eighths. T: How many eighths are shaded? S: 6 eighths. T: Work with a partner to write an addition and a multiplication sentence to describe the decomposition. S: is equal to T: What do these addition and multiplication sentences tell you? S: The shaded area didn’t change. It’s still the same amount. The number of pieces increased, but the size of the pieces got smaller. Adding together all of the smaller units equals the total of the larger units shaded. Multiplying also equals the total of the larger units shaded and is easier to write out! Problem 2: Draw an area model to represent the equivalence of two fractions, and express the equivalence as the sum and product of unit fractions.