More Nets, More Surface Area

5 min

Teacher Prep
Setup
Display the image of box for all to see. 1 minute to notice and wonder, followed by whole-class discussions.

Narrative

This is the first Math Talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.

This Math Talk focuses on multiplication of two whole numbers. It encourages students to observe the impact of adjusting a factor and to rely on the structure of base-ten numbers and the properties of operations to find products (MP7).

Each expression is designed to elicit slightly different reasoning. In explaining their strategies, students need to be precise in their word choice and use of language (MP6). While many ways of reasoning may emerge, it may not be feasible to discuss every strategy. Consider gathering only 2–3 different strategies per expression. As students explain their strategies, ask them how the factors impacted their approach.

Launch

This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.

Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and are to give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2 or 3, given limited time), and might ask questions such as "Who thought about it in a different way?" The teacher then records the responses for all to see, and might ask clarification questions about the strategies before revealing the next problem.

Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies and record and display their responses for all to see.
  • Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem. 

Keep all previous problems and work displayed throughout the talk.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Find the value of each product mentally.

  • 6⋅156 \boldcdot 15
  • 12⋅1512 \boldcdot 15
  • 6⋅456 \boldcdot 45
  • 13⋅4513 \boldcdot 45

Sample Response

  • 90. Sample reasoning: (6⋅10)+(6⋅5)=90(6\boldcdot 10) + (6\boldcdot 5) = 90
  • 180. Sample reasoning: Since the 6 from the first question doubled to 12, and the 15 stayed the same, the product doubles to 180. This is because there are twice as many groups of 15 as in the first question.
  • 270. Sample reasoning: Since the 6 is the same as the in the first question, and the 15 tripled to 45, the product triples to 270. This is because the number of groups stayed the same, but the amount in each group got three times as large.
  • 585. Sample reasoning: Since the 45 is the same as the previous question, we can double the 6 and the product to get 540. We need one more group of 45, and 540+45=585540+45 = 585.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to ‾\underline{\hspace{.5in}}’s strategy?”
  • “Do you agree or disagree? Why?”
  • “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 4.NBT.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
  • 4.NBT.B.5·Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Addressing
  • 6.G.2·Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
  • 6.G.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
  • 6.G.A.2·Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas <span class="math">\(V = l w h\)</span> and <span class="math">\(V = b h\)</span> to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.
  • 6.G.A.4·Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

30 min