4th Grade Unit 1:  Number Relationships 

Unit Driving Question 

How do Number relationships help us solve real-world problems?

Essential Questions 

1. How does place value play a part in number operations?

2. Why do we use multiplication and division?

3. How can we represent and solve real-world situations using all operations and unknowns?

Big Ideas 

1. Decomposing and composing numbers allows for flexible methods of computation when multiplying large numbers.
2. Decomposing and composing numbers allows for flexible methods of computation when dividing large numbers.
3. We can use a letter or symbol to represent an unknown quantity and use knowledge of operations to solve for the unknown.

Technology Resources

Launch Task 

1 Lesson 

Arrays and Hundreds Charts:  Three-act lesson in which students will model numbers using arrays.  They will identify patterns, mathematical properties, and number relationships through exploration of the hundreds chart. (link to lesson)

Big Ideas for Development Lessons 

5 Weeks (approximately 2 weeks for Big Ideas 1 & 2, 1 week for Big Idea 3) 

Big Idea 1: Decomposing and composing numbers allows for flexible methods of computation when multiplying large numbers.

Collaborative Engagement  

1. Multiplication Strategy Number Talk - This task consist of a 1 - 2 day discussion about how place value and multiples of ten play a part in multiplication.  Explanation - Partial Products

Key Resources  

1. Multiplication Using the Area Model - This lesson includes a video tutorial explaining the Area Model, as well as a ‘Walk the Room’ idea for students to practice this skill. (source:  betterlesson.com)

   Using the Area Model, Take Two - This lesson is a follow-up to the one above, and duplicates some of the original lesson.  However, one notable feature is a great idea at the end of the lesson for students to work collaboratively to show what they know.   (source:  betterlesson.com)
2. Multiplying by Multiples of Ten - This lesson provides additional practice for students, as well as requiring them to work collaboratively, discuss their work with peers and explain their thinking in writing.  (source:  betterlesson.com)
3. Multi-Step Problem Solving Using Multiplication - In this lesson, the teacher enlists the help of the Food Service Staff to present an application problem.  The situation presented will give students the opportunity to persevere in their work, and grapple with the mathematics a bit.  This lesson could easily be expanded to include additional problem-solving situations.  (source:  betterlesson.com)

Big Idea Formative Assessment 

1. Here is a link for an exit ticket to give after day 1 or 2 of exploring area model or partial product.
2. Here is a wonderful assessment to give at the conclusion of this Big Idea.  

Evidence of Understanding 

Justify that prior knowledge of basic facts can be used when multiplying larger numbers.

• Ex: Because 5 x 3 =15, generalize that 50 x 3 = 150   (15 x 10)

• 437 x 6 = (400 x 6) + (30 x 6) + (7 x 6)

Use estimation as a strategy to justify the reasonableness of products.

Solve multiplication problems (3 digit x 1 digit and 2 digit x 2 digit) using a variety of strategies and applying knowledge of basic facts.  

• Justify the product.

• Defend the validity of their strategy.

Use multiplication as a strategy to determine the number of items in a set, the number of equal groups, or the total number of items.

Justify strategies when solving real world problems.

• Compensation (using friendly numbers), partial product, cluster problems, area models, lattice, and standard algorithm.

• May solve a problem with division that is traditionally perceived as a multiplication problem.

Big Idea 2: Decomposing and composing numbers allows for flexible methods of computation when dividing large numbers.

Collaborative Engagement  

1. Sharing the Jelly Beans - This is a task in which students find various ways to share a bag of jelly beans fairly.  They will use prior knowledge about multiplication to devise strategies for dividing 2- and 3-digit numbers.  (Here is a link to an explanation of the partial quotients division strategy.)

Key Resources 

1. A Remainder of One - This lesson uses the book, A Remainder of One, to help students learn to interpret remainders.  They will also illustrate the situations presented in the book.   (source:  betterlesson.com)
2. Making Sense of Remainders - This lesson provides students with more opportunities to interpret remainders and includes additional problems for students to solve.  (source: betterlesson.com)
3. Real World Division Problems - Here are 5 division problems using real-world scenarios.  Depending on the students with whom you are working, these could be given to a pair or a group of students.  The first 3 problems have no remainder, but there will be remainders for problem 4 and 5.  Be sure to encourage students to explain how they solved the problems, as well as what the remainders mean for the final two problems.  (Answer key on p. 2)  

Big Idea Formative Assessment 

1. Here is an informal assessment regarding the relationship between multiplication and division.
2. Here is an exit ticket to give after 1 or 2 days of exploring division of multi-digit numbers.

Evidence of Understanding  

Justify that prior knowledge of basic facts can be used when dividing larger numbers.

• 140/7 = 20 because 14/7= 2 and multiply the 2 x 10 because the 14 is multiplied by 10 (140)

Describe the relationship between multiplication and division.

• Multiplication and division are inverse operations; each operation can be used to verify the accuracy of the other.

Solve division (2 or 3 digit dividend and 1 digit divisor) problems using a variety of strategies and applying knowledge of basic facts.

• Justify the quotient.

• Defend the validity of their strategy.

Use division as a strategy to determine the number of items in a set, the number of equal groups, or the total number of items.

Use estimation as a strategy to justify the reasonableness of quotients.

Justify strategies when solving real world problems.

• Partial quotient, partition or fair share model, standard algorithm.

• May solve a problem with multiplication that is traditionally perceived as a division problem.  

Big Idea 3: We can use a letter or symbol to represent an unknown quantity and use knowledge of operations to solve for the unknown.

Collaborative Engagement 

1. Horses in the Pasture - This task introduces students to the concept of using a variable to represent an unknown quantity.

Key Resources  

1. NCTM's Pan Balance - Description from NCTM site, ”Use this tool to strengthen understanding and computation of numerical expressions and equality. In understanding equality, one of the first things students must realize is that equality is a relationship, not an operation. Many students view "=" as "find the answer." For these students, it is difficult to understand equations such as 11 = 4 + 7 or 3 × 5 = 17 – 2.”
2. Real-World Problems - Solving for Unknowns - Four application problems based on real-world scenarios.  Students will devise the appropriate equation, focusing on using variables to represent unknown quantities.
3. Unknowns in Inequalities - In this lesson, students are introduced to finding values for unknowns in expressions involving the inequality symbols (< , >). 

Big Idea Formative Assessment 

1.  Here is an exit ticket for solving for unknowns.
2.  Here is an exit ticket for solving inequalities.   

Evidence of Understanding 

Solve for an unknown in equivalent and non-equivalent expressions.

• Use properties of operations

• Use the relationships between the operations (addition/subtraction, multiplication/division)

• Non-equivalent expressions may have multiple answers while equivalent expressions will have one answer or multiple answers with contingencies.

• 3 + f = 2 + r, contingency

Describe real-world situations using number sentences and vice versa.

• represent the unknown quantity with a letter or symbol.

Use comparison symbols to explain the relationships between numbers.

• The equal sign means ‘having equivalent value'.

Unit Closure

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

• Times Machine Performance Assessment- This performance assessment covers 4.N.1.1, 4.N.1.5, 4.N.1.7, 4.A.2.1, and 4.A.2.2. For access to scoring rubric and student samples, complete the NextThought PLC Common Assessment Discussion, Implementation, and Analysis training module.