PRINT VERSION - Multiplicative Reasoning

This unit will help students’ conceptual understanding of multiplication. Using real world examples and Unifix cubes (manipulatives), students will strengthen their understanding of multiplication.

Goals and Standards

---

Colorado Academic Standards:

• 3.OA.A. Operations & Algebraic Thinking: Represent and solve problems involving multiplication and division
  • 3.OA.A.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7.
  • 3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Mathematical Practice Standards: 

• Make sense of problems & persevere in solving them.
• Model with mathematics.
• Use appropriate tools strategically.
• Attend to precision.
• Look for & make use of structure.​

Learning Goals:

• Students can use groups to count how many objects they see and write multiplication equations to match.
• Students can label units and show their thinking with numbers, words and pictures.

Next Section:

Progression of Skills (outline)

Before the lesson:

• Mastery of “Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.” (2.OA.C.4) will be helpful, though not required, to make use of this intervention. 
  • Students should complete the following pre-assessment prior to entering this intervention. Multiplicative Reasoning - Pre-Assessment
  • Note: If a student can easily write both multiplication expressions that match the given array, then they may not need this intervention. If a student struggles with the pre-assessment provided, then this intervention will be more effective. 

During this toolkit lesson:

• Students can model multiplicative reasoning with numbers, words, and pictures. Students can understand multiplicative reasoning in context and include labels.

After the lesson:

• Students are ready to apply properties of operations as strategies to multiply and divide (3.OA.B.5).

Lesson

Directions for Intervention Activity:

Session 2 (With a partner)

• Session 2 will have students work in pairs to complete activity called: Please Go Bring Me (PGBM) Multiplicative Reasoning - "Promoting conceptual understanding of multiplication" article - Please Go Find Me.pdf

1. Student pairs.
   1. Students play PGBM in pairs. We recommend starting with pairs of students at different levels, because learning the game with tangible objects is considered available to most students (grade 3 or above). Later, the teacher can change pairs to differentiate instruction (e.g., pair students who both still need tangible cubes).
2. Student roles.
   1. In PGBM, students play one of two roles: Sender and Bringer. 
      1. Sender.
         1. Without telling the Bringer, the Sender thinks: “How many towers will I tell the Bringer to go and get? How many cubes will be in each tower?” Then, the Sender asks for the towers one at a time until the Bringer has brought all the needed towers. Next,  the Sender asks the Bringer Four Key Questions (See #3).
      2. Bringer.
         1. The Bringer brings towers to the Sender one at a time. She builds each tower from stackable, individual cubes. Once the Bringer has brought all requested towers, and while using complete sentences that explicate the units  considered, the Bringer answers the Four Key Questions. The Sender and Bringer compare answers and reasoning strategies. 
      3. Four Key Questions.
         1. After receiving all requested towers, the Sender asks Four Key Questions of the Bringer. These questions, and the teacher’s emphasis on always answering them by using complete sentences, support students’ reflection, so they explicitly distinguish and relate the three kinds of units. Teachers often displayed these Four Key Questions, as well as Sentence Starters for the Bringer’s response, on posters.
         2. The teacher should use the provided questions to encourage mathematical discourse: 
            • Q: How many towers did you bring?
              • I brought _______ towers.
            • Q: How many cubes are in each tower?
              • There are _______ cubes in each tower.
            • Q: How many cubes did you bring in all?
              • I brought _______ cubes in all.
            • Q: How did you figure this (total) out?
              • I figured out this total by _______.

Session 3 (Small group) 

• Session 3 will have students work in small groups to build different arrays with their cubes. 
• Each group will need their own supply of cubes and should build three or four separate arrays.
• Instruct students to build arrays with towers of equal cubes ranging from 1 to 9. This is an intervention intended to use multiplication and division within 100. 
• After student groups build their arrays, the challenge will be to have students label their array using ‘towers of cubes.’ Take the following two arrays:
  • (Array A)
  • (Array B)
• Possible addition equations to match the arrays could be. (4+4+4)  or (3+3+3+3). This is not what we are looking for. 
• By asking students to label ‘towers of cubes’ an appropriate response for (array a) would be, “This is an array of 3 towers with 4 cubes in each tower.” Represented as (3x4) 
• An appropriate response for (array b) would be, “This is an array with 4 towers with 3 cubes in each tower.” Represented as (4x3) 
• The provided document provides sentence frames and scaffolding to help children express their thinking. Multiplicative Reasoning - Session 3
• Asking students to label their arrays in this manner allows students to use multiplicative reasoning through multiplicative double counting. “The key to multiplicative double counting is that a student can be stopped in their counting sequence at any point and asked: How many towers have you counted? When asked, the student could tell you how many groups could be counted, as well as how many cubes per group and how many total cubes counted so far. This ability is not necessarily available to students who are simply using repeated addition, as they are likely not paying attention to the amount of groups they have counted as they are counting them.” How to Build Students’ Multiplicative Reasoning Skills – Digital Promise
• The teacher should use the provided questions to encourage mathematical discourse: 
  • How many cubes are in each tower? 
  • How many towers are in this array?  
  • How many total cubes are in this array? 
  • How can you show your thinking about this array using multiplication? 

• Multiplicative thinking is all around us. The real world application of the intervention is to have students think multiplicatively. That is to approach math problems by thinking about both units being presented in the problem, such as:
  • Legs vs Dogs
  • Legs vs Octopi 
  • Pieces of gum vs packages of gum 
  • Cubes vs towers
• This intervention also gives students different data and asks them to provide situations that have them use multiplicative reasoning. 

• Baker, L. R. (2018). Learning multiplication with puppies and Kittens. Enslow Publishing. 
• Fosnot, C. T., & Dolk, M. L. A. M. (2001). Young mathematicians at work. constructing number sense, addition, and subtraction. Heinemann. 
• How to Build Students’ Multiplicative Reasoning https://drive.google.com/file/d/1p9qysKih361ryyBc7Zz7fsnmuV7CBewm/viewSkills – Digital Promise  February 17, 2021 | By Nicola Hodkowski
• Tzur 2019_Elementary Thinking Path.pdf - Google Drive  -Ron Tzur