Lesson 10
Todo tipo de denominadores
Warmup: Cuántos ves: Suma de fracciones (10 minutes)
Narrative
The purpose of this How Many do You See is for students to visualize a common denominator for two fractions. The diagram can be seen as showing \(\frac{8}{12} + \frac{3}{12}\) but it can also be seen as showing \(\frac{2}{3} + \frac{1}{4}\). The area diagram provides a way to visualize why the product of two denominators works as a common denominator for two fractions.
Launch
 Groups of 2
 “¿Cuántos ven? ¿Cómo lo saben?, ¿qué ven?” // “How many do you see? How do you see them?”
 Display the image.
 1 minute: quiet think time
Activity
 Display the image.
 “Discutan con su compañero cómo pensaron” // “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Record responses.
Student Facing
¿Cuántos ves? ¿Cómo lo sabes?, ¿qué ves?
Student Response
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Activity Synthesis
 “¿Cómo se ve \(\frac{2}{3} + \frac{1}{4}\) en el diagrama?” // “How does the diagram show \(\frac{2}{3} + \frac{1}{4}\)?” (There are \(\frac{2}{3}\) of the left square and \(\frac{1}{4}\) of the right square.)
 “¿Cuál es el valor de \(\frac{2}{3} + \frac{1}{4}\)? ¿Cómo lo saben?” // “What is the value of \(\frac{2}{3} + \frac{1}{4}\)? How do you know?” (\(\frac{11}{12}\) because there are 11 shaded pieces and each one is \(\frac{1}{12}\).)
Activity 1: Denominadores diferentes (15 minutes)
Narrative
The purpose of this activity is for students to apply what they have learned about using common denominators to add and subtract fractions with unlike denominators. In a previous lesson, students added two fractions where neither denominator was a multiple of the other, \(\frac{1}{2} + \frac{1}{3}\), using a strategy that made sense to them. In this activity students see more complex examples. Having built an understanding that they need to find equivalent fractions with a common denominator students will develop strategies for finding a common denominator (MP7, MP8). Monitor for these students who:
 look at multiples of the denominators and pick a common one
 notice that the product of the denominators is a common denominator for the two fractions
Supports accessibility for: Conceptual Processing, Memory
Launch
 Groups of 2
Activity
 5 minutes: individual work time
 5 minutes: partner discussion
 monitor for students who:
 use twelfths as a common denominator to find the value of \(\frac{3}{4}+\frac{4}{6}\)
 use twentyfourths as a common denominator to find the value of \(\frac{3}{4}+\frac{4}{6}\)
Student Facing
Encuentra el valor de cada expresión. Explica o muestra cómo pensaste.
 \(\frac{3}{4}+\frac{7}{8}\)
 \(\frac{3}{4}+\frac{4}{6}\)
 \(\frac{3}{4}\frac{2}{5}\)
Student Response
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Advancing Student Thinking
If students don’t use equivalent fractions to find the value of the expressions, ask, “¿Cómo puedes escribir una suma (o una diferencia) equivalente usando fracciones que tengan el mismo denominador?” // “How can you write an equivalent sum (or difference) of fractions with the same denominator?”
Activity Synthesis
 Ask previously selected students to share their responses.
 “¿Cómo decidieron cuál denominador común usar?” // “How did you decide which common denominator to use?” (I know that 4 and 6 are both factors of 12 or I know that 4 and 6 are factors of 24 because 24 is \(4 \times 6\).)
 “¿Cómo usaron el denominador común para encontrar la suma?” // “How did you use the common denominator to find the sum?” (I found equivalent expressions with 12 or 24 as a denominator and then I could add the fractions since they had the same denominator.)
 “En la próxima actividad, vamos a ver una estrategia general para encontrar un denominador común de dos fracciones” // “In the next activity, we are going to see a general strategy to find a common denominator for two fractions.”
Activity 2: Multipliquemos los denominadores (20 minutes)
Narrative
The purpose of this activity is for students to explain why the product of the denominators of two fractions is always a common denominator for the two fractions. Students noticed in the previous activity that there are several possible common denominators. Sometimes it is possible to just see a common denominator. For example, for \(\frac{2}{3} + \frac{5}{9}\) students might notice that 9 is a common denominator because it is a multiple of 3. It can be convenient, however, to have a strategy that always works, especially for more challenging denominators. After explaining why the product of two denominators is always a common denominator for a pair of fractions (MP3), students practice finding sums and differences of fractions in any way that makes sense to them. This may include
 using the product of the denominators
 thinking about each pair of fractions individually
Both strategies are important. For example, \(\frac{3}{50} + \frac{11}{100} = \frac{17}{100}\) since \(\frac{3}{50}\) is equivalent to \(\frac{6}{100}\). The number\( \frac{17}{100}\) is probably easier to grasp mentally than the number \(\frac{850}{50,000}\)which is what you get if you use the product of the denominators.
Advances: Speaking
Launch
 Groups of 2
Activity
 5 minutes: individual work time
 5 minutes: partner work time
 Monitor for students who use different denominators for the last two sums and differences.
Student Facing
 Esta es la estrategia de Lin para encontrar el valor de \(\frac{2}{5} + \frac{4}{9}\): “Yo sé que \(5 \times 9\) es un denominador común, así que lo voy a usar”. ¿La estrategia de Lin para encontrar un denominador común funciona? Explica o muestra cómo pensaste. Después, encuentra el valor de \(\frac{2}{5} + \frac{4}{9}\).

Encuentra el valor de cada expresión usando un método que tenga sentido para ti.
 \(\frac{3}{8} + \frac{1}{5}\)
 \(\frac{7}{10}  \frac{2}{3}\)
 \(\frac{7}{20} + \frac{41}{50}\)
 \(\frac{2}{9}  \frac{1}{6}\)
Student Response
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Activity Synthesis
 Invite students to share how they found the value of \(\frac{3}{8} + \frac{1}{5}\)
 “Comparen su estrategia y el método de Lin. ¿Qué pueden decir?” // “How did your strategy compare to Lin’s method?” (I used the product of the denominators for a common denominator just like Lin.)
 Invite students to share how they found the value of \(\frac{7}{10}  \frac{2}{3}\).
 “¿La estrategia de Lin también funciona en este caso?” // “Does Lin’s strategy work here too?” (Yes, I used 30 which is \(10 \times 3\).)
 Invite selected students to share their responses for \(\frac{7}{20} + \frac{41}{50}\).
 Display: \(\frac{117}{100}\) and \(\frac{1,170}{1,000}\)
 “¿Estas fracciones son equivalentes? ¿Cómo lo saben?” // “Are these fractions equivalent? How do you know?” (Yes, the numerator and denominator for the second one are 10 times the numerator and denominator of the first.)
 “¿Cuál de estos denominadores prefieren?” // “Which of these denominators do you prefer?” (I like using hundredths because I’m used to them. I like using thousandths because I did not have to think about finding the common denominator. I just took the product of 20 and 50.)
Lesson Synthesis
Lesson Synthesis
“Hoy investigamos distintas formas de sumar y restar fracciones” // “Today we investigated different ways to add and subtract fractions.”
Display: \(\frac{2}{9}\frac{1}{6}\)
“¿Cómo podemos encontrar el valor de esta expresión?” // “How can we find the value of this expression?” (We can find a common denominator for the two fractions.)
“¿Qué denominadores comunes usaron?” // “What are some common denominators that you used?” (18, 36, 54)
“¿Qué observan acerca de estos denominadores comunes?” // “What do you notice about these common denominators?” (They are all multiples of 6. They are all multiples of 9. 36 is double 18 and 54 is triple 18.)
“¿Cuál denominador usaron para encontrar el valor de \(\frac{2}{9}\frac{1}{6}\)? ¿Por qué escogieron ese denominador?” // “Which denominator did you use to help you find the value of \(\frac{2}{9}\frac{1}{6}\)? Why did you choose that one?” (I chose 18 because it is the smallest. I chose 54 because I know that \(9\times6=54\).)
Cooldown: Suma de fracciones (5 minutes)
CoolDown
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