Unit 2 Number Relationships (Multiplicative Thinking) 

Unit Driving Question

How can we use multiplication strategies to solve real world problems?

Essential Questions

1. How can we represent numbers in different ways?

2. What patterns and relationships do we find in mathematics?

3. How do we apply multiplication and division strategies when problem solving?

4. How do we use properties of multiplication?

Big Ideas

1. Multiplication is the total number of items when given a number of equal groups and the number of items in each group.
2. The commutative, associative, and identity properties are used to find equivalent expressions.
3. Division is sharing a number into equal groups and finding the number of groups or the number of items in each group.
4. Number relationships determine the pattern, rule, or unknown number.

Technology Resources 

The following apps, websites, and smartboard lessons can be used throughout the unit, as needed, during small groups, lessons, to reinforce standards.  They are also useful for students who may need reinforcement, remediation, or differentiation.

• Third Grade List- This list has resources that can be used all year long.

Resources Within This Unit

• Big Idea 5:

• Two-Digit by One-Digit Online Game: This is an online game students can do to practice.

Additional Resources

• Interactive Multiplication Chart: This is an online multiplication chart where teachers and students can use different colors to highlight features on the chart. This could be used during a conversation on observations or reviewing specific facts.

• Llama Drama- This app would be good to use early in the unit when teaching multiplication strategies.

• Multiplication.com- This website has various games for students to play to help them learn their facts.

Launch Task

1 Lesson

Race Car Problem: This problem has students explore the idea of multiplication before any instruction. They must think of a strategy to solve the problem, such as drawing a picture, making a table, etc. All methods should be encouraged and explored!

Big Ideas for Development Lessons

5-6 Weeks

Big Idea 1: Multiplication is the total number of items when given a number of equal groups and the number of items in each group.

Collaborative Engagement 

1. Silverware Problem: Students can draw pictures, use manipulatives, or any other tool they may need to solve the problem.

Key Resources 

1. Multiplication Fluency: This is a great article that provides multiple ideas on how to build fluency with students.  Here are some posters that show some fact visuals. Here is another poster that shows all the multiplication strategies.
2. Array City -Students will make at least 3 buildings, using rulers to make straight lines, and write the multiplication facts.
3. Multiplication Word Problems: A freebie from Teachers Pay Teachers

Big Idea Formative Assessment

1. Football Game Formative Assessment: This task has multiple steps that students must pay attention to each step of the problem. Adapted from North Carolina 3-5 Formative Instructional and Assessment Task
2. Ways to Show: This is a math probe that has students evaluate if the representation fits the multiplication problem.

Evidence of Understanding 

• Models provide a concrete representation of multiplication.

• Division is the inverse operation of multiplication. 

Big Idea 2: The commutative, associative, and identity properties are used to find equivalent expressions.

Collaborative Engagement

1. Colored Pencils Task: This task has students working with three numbers that have to be multiplied. It challenges them to come up with a way to find the answer. Adapted from Illustrative Mathematics (original document link)

Key Resources 

1. Multiplication Foldable: A graphic organizer for students to make while learning the properties.
2. Properties of Multiplication: This resource includes many different ideas on how to practice the multiplication properties. It does include the distributive property, which could be removed from the lesson or activity.
3. Properties Sort: This is a sort of the properties, including the zero property.

Big Idea Formative Assessment 

1. Properties Formative Assessment: This is a formative assessment that can be used towards the end of this big idea.

Evidence of Understanding 

Apply properties to form a conceptual understanding of equivalent expressions.

• Flexible methods for computation require deep understanding of the operations and properties of operations.

Big Idea 3: Division is sharing a number into equal groups and finding the number of groups or the number of items in each group.

Collaborative Engagement 

1. Sharing Pencils: This task activates knowledge they already have and make them apply it in a new way. This will give teachers the opportunity to see how students can apply multiplication in a division context. Adapted from North Carolina 3-5 Formative Instructional and Assessment Task

Key Resources 

1. Division Strategies Poster. Fact families should also be thoroughly reviewed during this time so students get a concrete understanding of how multiplication relates to the division. Here are flashcards or fact family house template that can be used for students to fill out for a certain fact. 
2. Fair Tickets: This has students use multiplication strategies to help them solve division problems. Adapted from North Carolina 3-5 Formative Instructional and Assessment Task
3. Division Stories: Students will solve division word problems using manipulatives

Big Idea Formative Assessment

1. Field Trip Task: This task has students think of different ways to split students into field trip groups. Adapted from North Carolina 3-5 Formative Instructional and Assessment Task

Evidence of Understanding

Create representations of division by breaking down the total number into equal groups.

• Models provide a concrete representation of division.

• Multiplication and division are related as inverse operations.

• Strategies such as skip counting, repeated subtraction, and the inverse fluency of basic multiplication facts are tools used when solving problems. 

Big Idea 4: Number relationships determine the pattern, rule, or unknown number.

Collaborative Engagement (H3) 

1. Growing Patterns: Students will use geometric shapes to build patterns and number relationships.

Key Resources

1. Growing and Shrinking Patterns: This is a task from NCTM that has students explore growing patterns and engage in meaningful discussions. Here is the worksheet that goes with the lesson.
2. Shape Patterns: Students will generate shape patterns. Students will: create shape patterns, given a rule and description/drawing of the first element in the pattern. They will identify apparent features of shape patterns. The following books are also a great way to engage students and bring in a literacy connection: Pattern Book and Pattern Book 2 
3. Finding Unknown Values: This task assesses their knowledge of math facts as well as finding unknown values. The task can be changed to be more multiplication based or it can be used to review previous skills as well.
4. Network News: This can be used as an assessment of determining the unknown number (in & out box) using all four operations. 
5. Multiplication Rings: A task like this one shows how multiplication and division are related to one another.
6. Input/Output Tables: This website includes a lesson plan and materials that can be used to assess students’ understanding of input/output tables.
7. Guess My Rule: Function machine activity, using addition and multiplication. Here is a website you can use to practice as well: What is My Rule 

Big Idea Formative Assessment

1. Patterns Assessment: This can be used as a formative or summative assessment for both number and geometric patterns.
2. Shape Pattern Assessment: This assessment focuses on extending and analyzing shape patterns.

Evidence of Understanding (H3) 

Determine the pattern based on how the numbers relate to each other.

• Patterns can grow or shrink based on the rule.

• Patterns recur in multiple mathematical concepts.

• Patterns can consist of geometric shapes, numbers, and pictures. 

Analyze the numbers given in an input/output relationship to determine the rule and find missing values.

• The value of numbers and how they relate to one another can help determine the rule or unknown number when

• Problem-solving strategies for addition, subtraction, and multiplication provide guidance when analyzing input/output tables, function machines, and patterns. 

Big Idea 5: Place value strategies can be applied when multiplying two-digit by one-digit numbers.

Collaborative Engagement

1. Movie Word Problem: This problem has students try to use manipulatives such as money, pictures, etc. to solve the problem. 

Key Resources

1. Modeling Multiplication: This is a hands-on activity that models this multiplication skill.
2. Reason about the Product: This is a more difficult probe that includes estimation and multiplicative reasoning.
3. Two-Digit by One-Digit Online Game: This is an online game students can do to practice.

Big Idea Formative Assessment

1. Why or Why Not?: This formative assessment has them evaluate whether someone’s answer is correct and explain why or why not.

Evidence of Understanding 

Use place value strategies such as decomposing to multiply two-digit by one digit numbers.

• Decompose the two-digit number and multiply separately before combining the values. Example: 24x8…. 24 would be decomposed into 20 and 4. Then the students multiply 20x8 (160) and 4x8 (32). The two products are then added to find the total product (160+32= 192).

Unit Closure

1 Week (includes time for probes, re-engagement, and assessment)  

• Race Car Problem (Part 2)- This problem is similar to the initial task, but there is more rigor.
• Toys for Sale: Performance Assessment- Assessment over 3.N.2, 3.N.2.1, 3.N.2.2, 3.N.2.3, 3.N.2.5, 3.N.2.6, 3.N.2.7, 3.N.2.8. For access to scoring rubric and student samples, complete the NextThought PLC Common Assessment Discussion, Implementation, and Analysis training module.
• The Hexabug: Performance Assessment- Assessment over 3.A.1, 3.A.1.1, 3.A.1.2, 3.A.1.3, 3.N.2, 3.N.2.1. For access to scoring rubric and student samples, complete the NextThought PLC Common Assessment Discussion, Implementation, and Analysis training module