Lesson 11

What’s the Difference?

Warm-up: Number Talk: Subtracting Fractions (10 minutes)

Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for subtracting fractions with unlike denominators. These understandings help students develop fluency and will be helpful later in this lesson when they will need to be able to subtract fractions with unlike denominators.

Launch

  • Display one expression.
  • “Give me a signal when you have an answer and can explain how you got it.”
  • 1 minute: quiet think time

Activity

  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Facing

Find the value of each difference mentally.

  • \(\frac{2}{3}-\frac{1}{6}\)
  • \(\frac{2}{3}-\frac{1}{2}\)
  • \(\frac{2}{3}-\frac{4}{6}\)
  • \(\frac{2}{3}-\frac{1}{4}\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Display first two differences.
  • “How are the differences related?” (\(\frac{1}{6}\) and \(\frac{1}{2}\) are \(\frac{2}{3}\) so I can take away \(\frac{1}{6}\) and that gives me \(\frac{1}{2}\) or I can take away \(\frac{1}{2}\) and that gives me \(\frac{1}{6}\).)

Activity 1: Greatest Difference (15 minutes)

Narrative

The purpose of this activity is for students to practice subtracting fractions with unlike denominators. The structure of this activity is identical to the first activity in the previous lesson except that students are calculating differences instead of sums. Monitor for students who:

  • try to make one fraction in each pair as large as they can and the other fraction as small as they can
  • find a common denominator for all 4 differences in order to add them together

While playing the game, students may find that they have a smaller fraction as the minuend and a larger fraction as the subtrahend. There are several ways students could navigate this situation, one of which is to switch the order of the fractions.

MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Advances: Reading, Representing
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were most useful to solve the problem. Display the sentence frame, “The next time I subtract fractions, I will look for/pay attention to . . . ”
Supports accessibility for: Conceptual Processing, Attention, Memory

Required Materials

Materials to Gather

Required Preparation

  • Each group of 2 needs a paper clip.

Launch

  • Groups of 2
  • “Take a minute to read over the directions for Greatest Difference.”
  • 1 minute: quiet think time
  • “Play Greatest Difference with your partner.”

Activity

  • 10–12 minutes: partner work time

Student Facing

Use the directions to play Greatest Difference with a partner.

  1. Spin the spinner.
  2. Each player writes the number that was spun in an empty box for Round 1. Be sure your partner cannot see your paper.
  3. Once a number is written down, it cannot be changed.
  4. Continue spinning and writing numbers in the empty boxes until all 4 boxes have been filled.
  5. Find the difference.
  6. The person with the greatest difference wins the round.
  7. After all 4 rounds, the player who won the most rounds, wins the game. 
  8. If there is a tie, players add the differences from all 4 rounds and the highest total difference wins the game. 
Spinner. 8 equal parts. 4, 1, 5, 2, 3, 1, 6, 2.

Round 1

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}} \,- \, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\)

Round 2

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\,- \, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\)

Round 3

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, -\, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\)

Round 4

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\,- \, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “What strategies were helpful as you played Greatest Difference?” (I tried to make the first fraction in each pair as large as possible and the second fraction in each pair as small as possible.)
  • “What is the biggest difference possible between fractions in this game? How do you know?” (\(\frac{6}{1} - \frac{1}{6}\) since \(\frac{6}{1}\) is the biggest number and \(\frac{1}{6}\) is the smallest.)
  • “Did anyone get \(\frac{6}{1} - \frac{1}{6}\) as one of their fractions?” (Answers vary.)

Activity 2: What is the Smallest Difference? (20 minutes)

Narrative

The purpose of this activity is for students to practice subtracting fractions with unlike denominators. This activity has the same structure as the previous activity except that students are looking for the smallest difference rather than the largest difference and this time they are given all of the numbers at once rather than spinning them one at a time. Some strategies to monitor for include:

  • trying to get differences that are 0 using equivalent fractions
  • trying to make all of the fractions as small as possible, that is using a similar strategy to the sum game when they tried to get the smallest possible sum

While playing the game, students may find that they have a smaller fraction as the minuend and a larger fraction as the subtrahend. There are several ways students could navigate this situation, one of which is to switch the order of the fractions.

Launch

  • Groups of 2
  • “Now, let’s see how close you can get to a difference of 0 using a given set of numbers.”

Activity

  • 5 minutes: independent work time
  • “Compare your total with your partner to see who is closest to 0. Describe your strategy to your partner.”
  • 10 minutes: partner discussion

Student Facing

Use the numbers below to fill in the squares. Find each difference. Add the 2 differences together.

  • 1
  • 2
  • 2
  • 3
  • 4
  • 5
  • 6

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, - \,\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, = \)

\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, - \,\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}} \,= \)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • “What strategies were helpful as you tried to find the smallest difference?” (I tried to choose equivalent fractions so that I could get a difference of 0. I tried to choose small fractions so then the difference would also be small.)
  • “Did anyone choose numbers that made the second fraction larger than the first?” (Yes.)
  • “What did you do?” (I switched the order of the fractions so that I could subtract.)
  • “Was anyone able to get a total of 0?” (No. I could get some equivalent fractions with sixths, or fourths, thirds, or halves but there was nothing I could do with fifths. If I put the 5 in the numerator I can’t make an equivalent fraction.)

Lesson Synthesis

Lesson Synthesis

“Today, we played some games that involved adding and subtracting fractions. What advice would you give to someone who was learning how to add and subtract fractions with unlike denominators?” (When the denominators are small like the ones we worked on today, you can usually see a common denominator. Even if you have to use the product of the denominators, it's not bad for fractions like fifths and sixths since 5 times 6 is 30.)

Cool-down: Reflect on Subtracting Fractions (5 minutes)

Cool-Down

For access, consult one of our IM Certified Partners.