Lesson 12

The purpose of this estimation exploration is for students to reason about the size of a complex fraction sum with large denominators. Students can see that 1 is a good estimate because one fraction is small and the other is close to 1. In the synthesis they refine this estimate to explain why the value of the sum is a little larger than 1. 

• Groups of 2
• Display the expression.
• “What is an estimate that’s too high?” “Too low?” “About right?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “How do you know that the sum is greater than 1?” (\(\frac{17}{19}\) is \(\frac{2}{19}\) short of a whole. Since 17ths are bigger than 19ths, adding \(\frac{3}{17}\) makes it greater than 1.)

The purpose of this activity is for students to add and subtract fractions and estimate sums and differences of fractions using the context of a recipe. Students may have different responses and reasoning for the estimation questions. In both cases, they can calculate and compare fractions but they may have different thoughts about how these differences would affect the recipe or what exactly it means for the recipe to make “about \(1\frac{1}{2}\) cups.” In the synthesis, students discuss the reasonableness of the estimates and how to make precise calculations (MP6). When students relate their calculations to Priya's salad dressing they reason abstractly and quantitatively (MP2).

• Groups of 2
• “What kind of ingredients do you like to put in your salad?” (lettuce, cabbage, beans, seeds, beets, tomatoes, cheese)
• “What kinds of dressings do you put on your salad?” (homemade, Italian, blue cheese, tamari)

• 1–2 minutes: quiet think time
• 6–8 minutes: small-group work time
• Monitor for students who:
  • estimate to determine that Priya’s recipe will make about \(1\frac{1}{2}\) cups of dressing 
  • add \(\frac{3}{4} + \frac{1}{3}+\frac{1}{2}\) to determine the precise amount of dressing Priya’s recipe will make

• “If Priya borrows a tablespoon of olive oil from her neighbor and uses it to make dressing, will she be putting in more or less olive oil than the recipe calls for?” (\(\frac{1}{16}\) is smaller than \(\frac{1}{12}\) so she will be putting in less olive oil.)
• “Do you think 1 tablespoon is close enough?”
• Poll the class.
• “How might Priya’s decision to use 1 tablespoon of olive oil change the salad dressing?” (It won’t make a difference because the difference is so small. It might taste more lemony or more mustardy because there is not as much oil. It might affect the consistency of the dressing a little.)
• Ask previously selected students to share their estimates for the amount of salad dressing in the given order.
• “Why might Priya estimate that the recipe makes \(1\frac{1}{2}\) cups of salad dressing?” (\(\frac{3}{4}\) is \(\frac{1}{4}\) away from 1 and \(\frac{1}{3}\) is close to \(\frac{1}{4}\).)
• “Does the recipe make more or less than \(1\frac{1}{2}\) cups? How do you know?” (More because \(\frac{1}{3}\) is more than \(\frac{1}{4}\).)
• “How many cups does Priya’s recipe make? How do you know?” (\(1\frac{7}{12}\), I added \(\frac{1}{3}\), \(\frac{3}{4}\), and \(\frac{1}{2}\).)

The purpose of this activity is for students to solve multi-step problems involving the addition and subtraction of fractions with unlike denominators. Students work with both fractions and mixed numbers and can use strategies they have learned such as adding on to make a whole number. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

• Groups of 2
• “You and your partner will each choose a different problem to solve and then you will discuss your solutions.”

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion

• “How were the problems the same? How were they different?” (For both problems I had to add fractions first and then subtract that total from another number. There were mixed numbers in both problems.)
• “How did you use equivalent fractions to solve these problems?” (All the fractions we worked with had different denominators so we had to find equivalent fractions with the same denominators in order to add or subtract.)