Lesson and Assessments - Multiplicative Reasoning

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Summary, Goals
and Standards

Lesson
and Assessments

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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 Introduction and Pre-Assessment

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Students should complete the following pre-assessment on multiplication problems 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.

Directions for Lesson Introduction:

Session one will get students thinking conceptually about multiplication without actually using the word ‘multiplication’.

Session 1, Part 1 (Whole Group):

Today’s goal will be to activate students' background knowledge .First we introduce students to the idea of multiplication using real world examples that all students can relate to.
The following kids book provides wonderful visuals and support. If available it may be used as an interactive read-aloud to kick off this session.
Baker, L. R. (2018). Learning multiplication with puppies and Kittens. Enslow Publishing.
Our first goal is just to brainstorm. Instruct students to answer the following question using numbers, words and pictures. Allow time for students to work independently and show their thinking.
- How many legs does one dog have?
- How many legs do two dogs have
- How many legs will four dogs have?
- How many legs will six dogs have?
You may have students work this out in math notebooks or use the provided template: Multiplicative Reasoning - Dogs and Legs Template
The next step is very important. Students will go back to the questions and label their units. In this case dogs and legs. Here are some questions to guide instruction.
- In question one, how many total dogs were there? How many total legs were there? Label those units as dogs and legs. 1 dog, 4 legs.
- In question two, how many total dogs were there? How many total legs were there? Label those units as dogs and legs. 2 dogs, 8 legs
- In question three, how many total dogs were there? How many total legs were there? Label those units as dogs and legs. 4 dogs, 16 legs
- In question four, how many total dogs were there? How many total legs were there? Label those units as dogs and legs. 6 dogs, 24 legs
By engaging in this activity and by labeling both the total number of dogs and total number of legs, students will be engaging in an introduction to multiplicative reasoning, specifically to promote students, ‘multiplicative double counting.’ They will be simultaneously keeping track of the total number of dogs and the total number of legs. This idea of multiplicative reasoning will carry over throughout the remaining intervention sessions. (Multiplicative Reasoning: Part of the Development of Mathematical Reasoning)

Anticipate Misconceptions:

Skip counting is not multiplicative reasoning.
As students are introduced to multiplication it is easy to explain it by skip counting. Using the example of dogs we could say 1 dog has 4 legs so 2 dogs have 8 legs and 4 dogs have 16 legs.
This may be representing using multiplication: 14=4 or 24=8 or 44=16
A ratio table may show this thinking better:

Dogs	Legs
1	4
2	8
4	16

Multiplicative reasoning or specifically multiplicative double counting is different from just skip 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 dogs have you counted?” When asked, the student could tell you how many dogs that they have counted, as well as how many legs each ‘group of dogs’ had.
The goal being to think simultaneously about how many total dogs and how many total legs 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 dogs they have counted as they are counting the legs. To put in another way, multiplicative double counting is thinking about both units simultaneously. In this example that is the ‘groups of dogs’ and the ‘total number of legs in each group of dogs.’ This type of conceptual thinking is what this intervention will access in the students’ mind.

Session 1, Part 2 (Independent):

In Part 2 of Session 1, students use the manipulatives they will be using through the duration of the intervention — cubes.
If this is the first time students have done work with cubes, they should be allowed some independent exploratory time to build towers from their cubes. The teacher will actively roam the room and should ask the provided probing questions to encourage mathematical thinking questions about their designs, without explicitly talking about multiplication and arrays.
- “Can you show me one cube all by itself?”
- “Could you build me a tower using 5 cubes?”
- “Are all the towers you built made with cubes? Why or why not?”
- These questions will be provided. Students may also gallery walk various designs and ask questions of their own about different student designs.

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

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).
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?

Lesson Closure and Post-Assessment

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Directions for Lesson Closure (Session 4):

Session 4 will be a 2-part post-assessment: Multiplicative Reasoning - Post-Assessment
- Part one provides students with two arrays and requires them to express their thinking with multiplication equations.
- Part two provides written and pictorial data and challenges students to write their own problems to match the data. The emphasis will be on labeling the units in each problem.
The post-assessment may be used formatively or summatively.
- Formative: A teacher could roam as students complete these assessments to check for understanding and provide some assistance as needed. Alternatively, students could complete the assessment in small groups. Formative assessments could be reviewed with the class.
- Summative: Students would take the post-assessment independently. A teacher would record their scores at the end.

Real-World Applications and Project Ideas

A globe with tools such as a map and measuring cup surrounding its border.

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.

Resources

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