This Number Talk encourages students to use what they learned about products of a whole number and a fraction, the relationship between each pair of factors, and the structure in the expressions to mentally solve problems.
Students may write all the products as fractions, including products greater than 1. If everyone expresses the last three products only as , , and , then during the Activity Synthesis, ask if students could write whole-number or mixed-number equivalents for these fractions. The reasoning elicited here will be helpful later in the lesson when students decompose whole numbers in order to subtract fractional amounts.
Launch
Display one expression.
“Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “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 strategies.
Keep expressions and work displayed.
Repeat with each expression.
Encuentra mentalmente el valor de cada expresión.
Student Response
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Advancing Student Thinking
Activity Synthesis
“¿En cuáles expresiones puede ser útil usar 1 unidad —o 12 doceavos— para encontrar el producto? ¿Cómo?” // “In which expressions might it be helpful to use 1 whole—or 12 twelfths—to find the product? How?” (The first three. We know 4 groups of make 1 whole, so:
2 groups of make
6 groups of make
12 groups of make 3
“¿Por qué puede ser un poco más difícil pensar en la última expresión en términos de 12 doceavos?” // “Why might it be a little harder to think of the last expression in terms of 12 twelfths?” (The 30 in is not a factor or a multiple of 12.)
Consider asking:
“¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____'s reasoning in a different way?”
“¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
“¿Alguien pensó en la expresión de otra forma?” // “Did anyone approach the expression in a different way?”
“¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”
Activity 1
Standards Alignment
Building On
4.NF.B.3.a
Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
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.
In this activity, students use any strategy that makes sense to them to reason about subtraction of a fraction from a whole number. They begin by using an image to support their reasoning. Later, when no image is given, students may use a variety of ways to find differences. In the Activity Synthesis students share, explain, and relate different approaches for solving the problem (MP3).
At this point, students are not expected to decompose or rewrite numbers, using expressions and equations. They may perform the reasoning intuitively and informally. Later, they will formalize the different ways of recording the decomposition of numbers for subtraction. During the Activity Synthesis, the teacher records students’ thinking as drawings, equivalent fractions, or expressions, so it is visible to all students.
Monitor for and select to share in the Activity Synthesis students with the following approaches to solve the second problem about pouring water out of a 4-cup pitcher:
Remove or count back at a time, using a discrete drawing (for example, 1 square for each cup) or a number line. Sample response:
A number line from 0 to 4, partitioned into thirds, with 5 jumps of to the left of 4
Teacher records above each whole on the representation, rewrites as , and records: .
Remove or count back whole numbers first, and then thirds, using discrete drawings or a number line. Sample responses:
A number line from 0 to 4, partitioned into thirds between 2 and 3, with a jump from 4 to 3 and then another jump of to the left
Teacher rewrites as or and records: and Rewrites 3 as and records: .
Representation: Access for Perception. Ask students to identify correspondences between the image of the pitcher and number lines they have worked with in previous lessons. Invite students to use diagrams to solve the task. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Groups of 2
Display the image of the graduated pitcher.
“¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
1 minute: quiet think time
1 minute: partner discussion
“Ahora resolvamos problemas en los que se sirve una bebida de una jarra” // “Let’s now solve some problems about pouring a drink out of a pitcher.”
Activity
7–10 minutes: independent work time
2 minutes: partner discussion
As you monitor for the approaches listed in the Activity Narrative, consider asking:
“¿En qué parte de su diagrama o de su recta numérica esta representada esta fracción?” // “Where is this fraction represented in your diagram or number line?”
“¿Cómo usaron lo que saben sobre fracciones equivalentes o descomposición de fracciones?” // “How did you use what you know about equivalent fractions or decomposing fractions?”
Una jarra contiene jugo de sandía. Si sirviéramos las siguientes cantidades, ¿cuántas tazas de jugo quedarían en la jarra? En cada caso, la jarra contiene 3 tazas de jugo de sandía al comienzo.
de taza
tazas
tazas
tazas
Otra jarra contiene agua. Si sirviéramos las siguientes cantidades, ¿cuántas tazas de jugo quedarían en la jarra? En cada caso, la jarra contiene 4 tazas de agua al comienzo.
Explica o muestra tu razonamiento. Si te ayuda, usa diagramas o ecuaciones.
de taza
tazas
tazas
Activity Synthesis
Invite previously selected students to share in the given order. Record or display their work for all to see.
Connect students’ approaches by asking:
“¿De qué formas pensaron en las 4 tazas enteras cuando restaron?” // “What were different ways to think about 4 whole cups when subtracting?” (3 cups and cup, 2 cups and cups, cups)
Connect students’ approaches to the learning goal by asking:
“¿Cómo se usaron las fracciones equivalentes en esas estrategias?” // “How did the strategies use equivalent fractions?” (We could think about each whole as or , or all the wholes completely partitioned into fourths or thirds on a diagram or number line or while explaining our thinking.)
Activity 2
Standards Alignment
Building On
Addressing
4.NF.B.4.c
Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
This activity makes explicit what students may have noticed in earlier activities, namely, that we can subtract a fraction from a whole number if we:
Write an equivalent fraction for the whole number.
Decompose the whole number into a sum of smaller numbers, which could be fractions, or a whole number and a fraction.
This sorting task gives students opportunities to analyze fractions and expressions closely and make connections (MP7). Each card has a value equivalent to either or . Though students may sort the cards in a few valid ways, the discussion should focus on the values on the cards. Students then apply their insights from the sorting work to subtract a fraction from whole numbers.
This activity uses MLR2 Collect and Display. Advances: conversing, reading, writing.
Launch
Groups of 2 or 4
Give each group a set of cards.
Activity
“Con su grupo, clasifiquen estas tarjetas en 2 categorías que tengan sentido para ustedes. En grupo, expliquen cómo razonaron” //“Work with your group to sort these cards into 2 categories in a way that makes sense to you. Work with your partner to explain your reasoning.”
5 minutes: group work time on the first problem
Monitor for students who sort the cards by:
The number of whole numbers they show.
Whether they show only a number or an expression.
Their value or by their equivalence to and .
MLR2 Collect and Display
Circulate to listen for and collect the language students use to describe the features of the expressions on the cards or the connections between expressions. Listen for terms such as “equivalent,” or “equivalent fractions,” “equal,” “sum,” “difference,” and “decompose.”
Record students’ words and phrases on a display, and update it throughout the lesson.
Pause before students proceed to the next problem. Invite 2–3 previously selected groups to share their sorting decisions, ending with a group that sorted the cards by value.
“Veamos esa última forma de clasificar. Las tarjetas de cada grupo tienen el mismo valor y ese valor es el resultado de restarle una fracción a un número entero” // “Let’s look at that last way of sorting. The cards in each group have the same value, and that value is a result of subtracting a whole number by a fraction.”
Display the cards as shown:
“Hablen con un compañero sobre cómo se relaciona cada tarjeta con la que está encima o con la que está debajo. Hagan esto con todos los grupos de tarjetas” // “Turn to a partner. Talk about how each card is related to the one above or below it. Do this for each group of cards.”
2 minutes: partner discussion
Invite 2–3 students to share the connections between the expressions in each group. Highlight that:
In the first group, the 1 can be written as an equivalent fraction, , which is helpful for subtracting .
In the second group, the 2 can be rewritten as an equivalent fraction, , or decomposed into a sum of , which is equivalent to . Both strategies help us to subtract .
5 minutes: independent work on the second problem
Activity Synthesis
See the Lesson Synthesis.
Lesson Synthesis
“Hoy aprendimos que podemos restarle una fracción a un número entero si reescribimos el número entero como una fracción o si descomponemos el número entero” // “Today we learned that we can subtract a fraction from a whole number by either rewriting the whole number as a fraction, or by decomposing the whole number.”
Select students to share how they found the three differences in the last problem in the last activity. As students explain, update the display by adding or replacing language or annotations.
“¿Qué otras palabras o frases importantes deberíamos incluir en nuestra presentación?” // “Are there any other words or phrases that are important to include on our display?”
Highlight explanations that show:
The 1 in can be written as .
The 2 in can be written as or decomposed into .
The 3 in can be written as or decomposed into (among other sums).
Writing an equivalent fraction and decomposing the whole number each make it easier to find the difference.
Remind students to borrow language from the display as needed in future activities.
Standards Alignment
Building On
4.NF.B.4.b
Understand a multiple of as a multiple of , and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express as , recognizing this product as . (In general, )
