# Discussing Scaling with Fractions

Challenge students' understanding of multiplying fractions by whole numbers to help them form opinions about rules regarding scaling. Use this lesson on its own or as support for the lesson Multiplication as Scaling.
This lesson can be used as a pre-lesson for the Multiplication as ScalingLesson plan.

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This lesson can be used as a pre-lesson for the Multiplication as ScalingLesson plan.

Students will be able to discuss multiplication as scaling.

##### Language

Students will be able to express an opinion about scaling with fractions using sentence frames and collaborative conversations.

(5 minutes)
• Display three fraction expressions with two numbers on the board where there is one factor in common between all the expressions (e.g., 1/3 x 5/3, 2/3 x 1/3, 3 x 1/3). Ask students to look at the fractions and tell you which answer will be more than one without solving the problem on paper.
• Allow students to think on their own for 30 seconds before turning to their partners to discuss the answer.
• Tell students that today they'll discuss how to estimate the product of a multiplication problem when the factors are whole numbers and fractions.
(6 minutes)
• Explain that ScalingIs when we predict the size of the ProductIn a multiplication problem based on the size of the two factors. We can EstimateThe answer to a multiplication problem involving fractions by considering the values of the factors. In the case of a multiplication expression involving a fraction, when we estimate we determine if the product will be Greater thanOr Less thanOne of the factors. For example, say, "In 1/2 x 2, I estimate the product will be 1 or greater than 1 because the factor 2 is greater than 1."
• Have students use their whiteboards to copy your teacher markings as you model solving 1/2 x 2With area models or the standard algorithm to support your estimation.
• Provide another estimate based on 8/2 x 1/2And then model verifying your answer to check your estimate.
• Define "always true" as something that always happens, "sometimes true" as something that sometimes happens but not all the time, and "never true" as something that never happens. Provide an everyday example to emphasize the meanings. For example, say, "When I get home from school, I always let my dog out of the kennel and give her water. Sometimes, we go for a walk, but not on days when it's raining. I never just leave my dog in the kennel when I come home from school."
• Present the following statement and ask students to determine if it is Always true, Sometimes true, or Never true: "If you multiply a fraction by 1, the answer will be less than 1."
(10 minutes)
• Have students consider the statement quietly for a minute and then let them share their thoughts with their partners. Tell them to use their whiteboards to take notes.
• Encourage students to provide examples of equations that would support their answer (e.g., 1 x 3/4 = 3/4).
• Write some key phrases on the board as they speak to give support to other students when they want to share. For example, "I think the answer is always true because any number multiplied by itself is another number," or "I think it's sometimes true because if the fraction is an improper fraction, like 1 x 6/2 = 6/2 = 3, then the answer could still be more than 1."
• Pass out a chart paper to each group of 3–4 students you form and hand out one square from the Always, Sometimes, and Never: Multiplication Scaling worksheet to each group. Tell students to determine if the statement on the cards is sometimes true, always true, or never true. Remind them to be ready to give a reason to support their answer with examples of expressions they create on their chart paper.
• Provide struggling groups multiplication expressions for them to consider and solve to help them decide about their statement. Also, give students sentence frames to justify their choices within the groups, such as "I know this because ____."
(10 minutes)
• Have a representative from each group present the group's statement and answer, along with their justification.
• Display the T-Chart with Three Columns worksheet with headings "Always True," "Sometimes True," or "Never True" in each column and place the cards from the teacher copy of the worksheet Always, Sometimes, and Never: Multiplication Scaling into the column students choose as each group presents their findings.
• Encourage observers to challenge the group's choices with examples of expressions of their own. As students present their findings, look for ways to extract the following rules from their statements and presentations:
1. When you multiply a whole number by a fraction LessThan 1, the answer will be LessThan the whole number.
2. When you multiply a whole number by a fraction GreaterThan 1, the answer will be GreaterThan the whole number.
3. When you multiply a whole number by a fraction EqualTo 1, the answer will be EqualTo the whole number.
• Write the above rules on a chart paper to serve as an anchor chart by the end of their presentations. Present the rules and ask for students to turn and talk to their elbow partner to see if they can reject one of the rules. Then, ask them to create an expression to support the rules if time allows.

Beginning

• Allow students to use their home language (L1) or their new language (L2) in all discussions. Provide bilingual reference materials to assist in their vocabulary word acquisition.
• Encourage them to use the vocabulary cards and terms in their conversations and writing. Allow them to draw pictures to support their understanding of the terms.
• Give them the option to share their ideas in partners before sharing them with the class. Provide sentence frames and word banks for their support as necessary.

• Pair students with mixed-ability groups so they can offer explanations and provide feedback to beginning ELs when appropriate.
• Ask them to present the information to the class but allow them to support other ELs as they respond to questions and challenges from the class.
• Have them write down potential sentence frames students can use with their justifications.
• Ask them to define key terms for the class, such as "improper fraction" or "mixed number."
(6 minutes)
• Distribute a blank sheet of paper and have students fold it once so that there are two boxes.
• Display the exercise Interpret Multiplication of Fractions as Scaling and ask students to write down two expressions, one in each box. Underneath each expression, ask them to determine which rule it applies to from the anchor chart, or if it challenged any rule.
• Provide a sentence frame, such as "This problem follows rule ____Because ____," and "The problem supports/challenges rule ____Because ____."
• Allow them to solve the expression after they've written their sentence to check their answer.
(3 minutes)
• Display the teacher copy of the T-Chart with Three Columns with the statements in their appropriate column and ask students to create their own "Always True" statement and share it with their elbow partner. Monitor student conversations and choose one to share with the class.
• Write the student statement on the board and have students put their thumbs up or thumbs down if they agree with the statement. If they disagree, ask them to share their rationale with the class.

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