The purpose of this Warm-up is to draw students’ attention to inequality statements. It reminds them of the meaning of inequality symbols and how to read the statements, which will be useful when students compare fractions later in the lesson. The Warm-up also elicits observations that an equation or inequality can be true or false. While students may notice and wonder many things, observations about comparison and about the meaning of the symbols and statements are the important discussion points.
Launch
Groups of 2
Display the four statements.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
Activity
“Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
1 minute: partner discussion
Share and record responses.
¿Qué observas? ¿Qué te preguntas?
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿Qué dice cada afirmación?” // “What does each statement say?”
If needed, guide the class to read each statement aloud.
“¿Cuáles de estas afirmaciones son verdaderas? ¿Cuáles no?” // “Which of these statements are true? Which ones are not?” (The first and last are true. The second and third are false.)
“¿Por qué son falsas?” // “Why are they false?” ( is equal to, not greater than, 4 wholes and . Four is greater than .)
Activity 1
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
The purpose of this activity is for students to compare two fractions and identify which is greater in ways that make sense to them. Previously, students classified fractions based on their relationship to and 1 (whether they are less than or greater than these benchmarks). Students used these classifications to compare fractions. In this activity, students are presented with fractions that are in the same group (for example, both less than , or both greater than but less than 1), so students need to reason in other ways to make comparisons.
Monitor for and select students with the following approaches to compare fractions in Group 2 to share in the Activity Synthesis:
Draw number lines or tape diagrams and reason informally about equivalent fractions.
Use the distance of each fraction from 0, , or 1 to reason about the relative size.
Reason about equivalent fractions numerically by writing out the multiplication.
The approaches are sequenced from more concrete to more abstract to support students as they compare fractions. Although students may compare in a number of ways, focus the discussion on the ways students relied on the idea of equivalence. Aim to elicit both key mathematical ideas and a variety of student voices, especially students who haven’t shared recently. For an example for each approach, see Student Responses.
Representation: Internalize Comprehension. Activate background knowledge. Invite students to review the strategies they know for comparing fractions (reasoning about denominators or numerators, comparing to a benchmark, and writing equivalent fractions). Record students’ strategies on a visible display, including details (words or pictures) that will help them remember how to use the strategy. Supports accessibility for: Conceptual Processing, Memory, Attention
Launch
Groups of 2
“Estas son unas fracciones que ya habíamos organizado en una lección anterior. Las comparamos con y con 1.” // "Here are some fractions you’ve sorted in an earlier lesson. We compared them to and 1.”
“¿Qué tienen en común las fracciones del grupo 3? ¿Por qué creen que están en el mismo grupo?” // “What do the fractions in Group 3 have in common? Why might they be in the same group?” (They are all greater than 1.)
“¿En qué son diferentes las fracciones del grupo 1 a las fracciones del grupo 2?” // “How are the fractions in Group 1 different than those in Group 2?” (Those in Group 1 are less than , and those in Group 2 are greater than but less than 1.)
“Podemos darnos cuenta de que las fracciones del grupo 2 son mayores que las del grupo 1 y que las fracciones del grupo 3 son mayores que las de los otros dos grupos” // “We can tell that the fractions in Group 2 are greater than those in Group 1, and the fractions in Group 3 are greater than those in the other groups.”
“Ahora comparen las fracciones dentro de cada grupo” // “Now compare the fractions in each group.”
Activity Synthesis
Invite previously selected students to share their reasoning about fractions in Group 2 in the given order. Record or display their work for all to see.
Connect students’ approaches by asking:
“¿En qué se parecen las formas de comparar las fracciones?” // “How are the ways of comparing the fractions the same?” (We had to find a way to deal with the different denominators.)
“¿En qué son diferentes?” // “How are they different?” (Some ways involved thinking about how close or how far away from 1 each fraction is. Other ways involved reasoning with equivalent fractions, and some showed the multiplication to write equivalent fractions.)
Connect students’ approaches to the learning goal by asking:
“Al comparar fracciones que no tienen el mismo denominador, ¿cómo les ayuda saber fracciones equivalentes?” // “How does understanding equivalent fractions help you compare fractions that don’t have the same denominator?” (We could change one fraction so it had the same denominator as the other fraction. Then it was easier to compare them.)
Activity 2
Standards Alignment
Building On
Addressing
4.NF.A.1
Explain why a fraction is equivalent to a fraction by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
The purpose of this activity is for students to compare pairs of fractions by writing one or more equivalent fractions. In all pairs of fractions given here, one denominator is a factor or a multiple of the other, which encourages students to convert one into an equivalent fraction with the same denominator as the other fraction. On repeated reasoning, students see that writing an equivalent fraction can facilitate the comparison (though in some cases, students may still find it efficient to reason in other ways).
This is the first time in IM Grade 4 that students use the symbols and to express comparisons, so some supports for reading aloud inequality statements are suggested in the Launch.
Launch
Groups of 2
Read together the four statements in the first question.
Consider writing out in words the meaning of the symbols and (“es mayor que” // “is greater than” and “es menor que” // “is less than”) and display them for students’ reference.
Activity
7–8 minutes: independent work time
2–3 minutes: partner discussion
Monitor for students who make comparisons by:
Using the relationship and distance to benchmark numbers.
Writing an equivalent fraction either by dividing or multiplying the numerator and denominator by the same number.
En cada caso, decide si la afirmación es verdadera o falsa. Prepárate para explicar o mostrar cómo lo sabes.
Compara cada pareja de fracciones. Usa los símbolos , o para hacer que cada afirmación sea verdadera.
Activity Synthesis
Select students to share their responses and how they reasoned about the comparisons.
Lesson Synthesis
“Hoy comparamos fracciones escribiendo fracciones equivalentes y usando otras estrategias” // “Today we compared fractions by writing equivalent fractions and by using some other ways.”
Ask students to find an example of a pair of fractions in today’s activity that it was helpful to compare by:
Reasoning about the denominators and numerators.
Seeing where the fractions are in relation to , 1, or another benchmark.
Writing an equivalent fraction for one of the fractions.
Standards Alignment
Building On
Addressing
4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
“Tómense unos minutos en silencio para trabajar en los problemas. Después, compartan sus respuestas con su pareja” // “Take a few quiet minutes to work on the problems. Afterward, share your responses with your partner.”
5–7 minutes: independent work time
3–5 minutes: partner discussion
As you monitor for the approaches listed in the Activity Narrative, consider asking:
“¿En qué pensaron para poder comparar estas fracciones que tienen denominadores diferentes?” // “How did you think about these fractions with different denominators to be able to compare them?”
“¿Cómo les ayudó en su razonamiento pensar en fracciones de referencia o fracciones equivalentes?” // “How did thinking about benchmark fractions or equivalent fractions support your reasoning?”
Estas son unas parejas de fracciones que se organizaron en 3 grupos. Marca la fracción mayor en cada pareja. Explica o muestra tu razonamiento.
Grupo 1:
o
o
Grupo 2:
o
o
Grupo 3:
o
o
Student Response
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Advancing Student Thinking
If students compare the numerators without considering that the denominators are different, consider asking:
“¿Cómo decidiste que ______ es mayor que ______? ” // “How did you decide that ______ is greater than ______?”
“¿Qué observas acerca de los denominadores? Dime una forma de partir las fracciones para que tengan el mismo número de partes en total” // “What do you notice about the denominators? What is a way to partition the fractions so that they have the same number of total parts?”
Student Response
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Advancing Student Thinking
4.NF.A.2
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols , =, or , and justify the conclusions, e.g., by using a visual fraction model.
