This Warm-up prompts students to carefully analyze subtraction expressions containing two fractions or a whole number and a fraction. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminology students use to talk about these expressions. To compare the values of the expressions, students need to perform subtraction and apply their knowledge of equivalence. The reasoning here also will be helpful later as students reason about differences of two mixed numbers or a mixed number and a fraction.
Launch
Groups of 2
Display the expressions.
“Escojan 3 expresiones que vayan juntas. Prepárense para compartir por qué van juntas” // “Pick 3 expressions that go together. Be ready to share why they go together.”
1 minute: quiet think time
Activity
“Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
2–3 minutes: partner discussion
Share and record responses.
¿Cuáles 3 van juntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Por qué las 4 van juntas?” // “Why do all 4 go together?” (All are subtraction expressions that have fractions, whole numbers, or mixed numbers. We had to find their values to compare them.)
“Cuando compararon las expresiones, ¿qué estrategias usaron para encontrar su valores?” // “As you compared the expressions, what strategies did you use to find their values?”
Activity 1
Standards Alignment
Building On
Addressing
4.NF.B.3.d
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
In this activity, students solve contextual problems that involve subtracting fractions of which at least one value is a mixed number and it is necessary to decompose one or both numbers. Students find differences in any way that makes sense to them. They may use number-line diagrams, reason in terms of addition, or perform repeated partial subtractions (without necessarily writing expressions or equations). For instance, to find the difference between and , they may:
Draw a number line, partition it into eighths, locate , and count back 7 spaces to the left to represent subtraction of .
Reason: “What number plus gives or ?”
First subtract from , and then subtract another .
First subtract 1 from , and then add back .
Students also may use the insights they gained previously, about rewriting and decomposing whole numbers, to facilitate subtraction and consider whether they could be applied to subtraction of a fraction from a mixed number. When students recognize the mathematical features of objects in the real world and ask questions that arise from a presented situation, they reason abstractly and quantitatively (MP2).
Action and Expression: Develop Expression and Communication. Provide access to a variety of tools, including blank pre-formatted tape diagrams, blank pre-formatted number lines, and colored pencils. Invite students to estimate or solve without these first, and then use them to check their work. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing, Social-Emotional Functioning
Launch
Groups of 2
Consider asking: “¿Qué cosas han hecho, dado o recibido para celebrar sus amistades?” // “What are some things you have made, given, or received that celebrate your friendships?”
Explain that macramé (MAA-kruh-may), a way of making textiles by knotting, is at least a few thousand years old. The name came from the Arabic word “miqramah” (mee-KRAH-mah), with one meaning being “fleco ornamental o decorativo” // “ornamental or decorative fringe.”
“Las pulseras de macramé son una forma popular de celebrar amistades. ¿Han visto o hecho una pulsera de la amistad?” // “Macramé bracelets are a popular way to celebrate friendships. Have you seen or made a friendship bracelet?” (If possible, prepare a couple of bracelets—finished and unfinished—for students to see, or show images or a video of a bracelet being made.)
“Usemos lo que sabemos sobre restar fracciones para resolver algunos problemas sobre pulseras de la amistad” // “Let’s use what we know about subtracting fractions to solve some problems about friendship bracelets.”
Activity
“En silencio, trabajen durante 5 minutos. Luego, discutan sus ideas y completen el resto con su compañero” // “Take 5 quiet minutes to work on the task, and then discuss your thinking and complete the rest with your partner.”
5 minutes: independent work time
5 minutes: partner discussion and work time
Monitor for the different ways students go about finding differences. Identify those who write subtraction or addition expressions or equations.
Clare, Elena y Andre hacen pulseras de la amistad tejidas en macramé. Quieren que sus pulseras midan pulgadas de largo. En cada pregunta, explica o muestra tu razonamiento.
Clare comienza primero a hacer su pulsera. Solo le faltan de pulgada para terminarla. ¿Cuál es el largo de su pulsera en este momento?
En este momento, la pulsera de Elena mide pulgadas de largo y la pulsera de Andre mide pulgadas de largo. ¿Cuántas pulgadas más le faltan a cada uno para llegar a pulgadas?
¿Cuánto más larga es la pulsera de Elena que la de Andre en este momento?
Activity Synthesis
Select 2–3 students who use different methods to briefly share their responses. Record subtraction and addition expressions or equations that students wrote or mentioned.
Students will take a closer look at the same numbers in the next activity.
Activity 2
Standards Alignment
Building On
Addressing
4.NF.B.3.c
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
Previously, students found differences of two fractions, including mixed numbers, using any way that made sense to them. This activity formalizes and makes explicit how such differences can be found by writing equivalent fractions and decomposing a whole number or a mixed number. When students share their responses with a partner and revise them based on the feedback they receive, they construct viable arguments and critique the reasoning of others (MP3).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: reading, writing.
Launch
Groups of 2
Display the four expressions in the Task Statement.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
30 seconds: quiet think time
Share responses. If not mentioned by students, point out that finding the values of these expressions is a way to answer the questions in the first activity.
“Estudiemos más a fondo cómo podemos restar fracciones como estas sin usar diagramas ni un contexto que nos ayuden” // “Let’s take a closer look at how we can subtract fractions like these, without using diagrams or a context to help us.”
Activity
“En silencio, analicen el primer grupo de cálculos durante 2 minutos. Luego, hablen con su compañero sobre lo que piensan que ocurre en los cálculos” // “Take 2 quiet minutes to study the first set of calculations. Then talk to your partner about what you think is happening in the calculations.”
2 minutes: quiet think time
1–2 minutes: partner discussion
Share responses.
“¿Por qué piensan que aparece descompuesto en distintas sumas?” // “Why do you think is decomposed into different sums?” (See Student Responses.)
“¿Qué muestran las últimas dos expresiones?” // “What do the last two expressions show?” (The first expression shows the being subtracted from the , which is a part of . The second expression shows the result of that subtraction being rejoined with the 8 from the .)
“Ahora es su turno de restar números mixtos. Si es necesario, descompónganlos” // “Now it’s your turn to subtract some mixed numbers, by decomposing them, if needed.”
“Con su pareja, completen los tres cálculos incompletos del segundo problema” // “Work with your partner to complete the three incomplete calculations in the second problem.”
5–7 minutes: partner work time
Estas son 4 expresiones que puede que hayas escrito para las pulseras de la amistad.
Esta es una forma de encontrar el valor de la primera expresión. Examina de cerca los cálculos. Habla con tu compañero sobre por qué está escrito como diferentes sumas.
Subtract. 9 and 4 eighths minus 7 eighths. Under first number. Line 1, 9 and 4 eighths. Line 2, 8 plus 1 plus 4 eighths. Line 3, 8 plus 8 eighths plus 4 eighths. Line 4, 8 plus 12 eighths. Under second number, 7 eighths. Bottom section. Line 1, 8 plus 12 eighths minus 7 eighths. Line 2, 8 plus 5 eighths. Line 3, 8 and 5 eighths.
Estos son unos cálculos sin terminar. Complétalos para encontrar el valor de cada diferencia.
a
Subtract. 9 and 4 eighths minus 5 and 1 eighth. Under first number. Line 1, 9 and 4 eighths. Line 2, 9 plus 4 eighths. Under second number. Line 1, 5 and 1 eighth. Line 2, 5 plus 1 eighth.
b
Subtract. 9 and 4 eighths minus 3 and 5 eighths. Under first number. Line 1, 9 and 4 eighths. Line 2, 8 plus 1 plus 4 eighths. Line 3, 8 plus 8 eighths plus 4 eighths. Line 4, 8 plus 12 eighths. Under second number. Line 1, 3 and 5 eighths. Line 2, 3 plus 5 eighths.
c
Subtract. 5 and 1 eighth minus 3 and 5 eighths. Under first number. Line 1, 5 and 1 eighth. Line 2, 5 plus 1 eighth. Under second number. Line 1, 3 and 5 eighths. Line 2, 3 plus 5 eighths.
Student Response
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Advancing Student Thinking
Activity Synthesis
MLR1 Stronger and Clearer Each Time
Ask each student to find a new partner.
“Explíquenle a su nuevo compañero su respuesta a la última parte del segundo problema” // “Explain your response to the last part of the second problem to your new partner.”
“Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta el momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
3–4 minutes: structured partner discussion.
Repeat with at least one other partner.
“Ajusten su respuesta inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial response based on the feedback you got from your partners.”
2 minutes: independent work time
Lesson Synthesis
Arrange students in groups of 4. Give each group tools for creating a display.
Assign to each group one expression from the second problem of the last activity. Ask them to record on a display the calculations for finding the value of that expression.
Invite groups to share their displays for all to see. Ask students to analyze others’ calculations and notice how they are like or unlike their own calculations.
Select a student display for each expression or display the calculations from the Student Responses. Ask students to identify places where mixed numbers are decomposed or equivalent fractions are written to facilitate subtraction.
“¿En qué casos puede ser útil o necesario descomponer un número entero en una suma cuando estamos restando una fracción?” // “When might it be helpful or necessary to decompose a whole number into a sum in order to subtract a fraction?” (When we are subtracting a fraction from a whole number or a mixed number and there aren’t any or enough fractional parts from which to subtract.)
Standards Alignment
Building On
Addressing
4.NF.B.3.c
Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
