Unit Title

Incorporating problem based learning when using operational and rational numbers


4 final project

Designed by

Bettina Meyer

Time Frame

Varies depending on students ( 3 + weeks)

Stage 1- Desired Results

Establish Goals

  • 5.8.2 Apply and adapt a variety of appropriate strategies to solve problems.
  • 5.8.3 Monitor and reflect on the process of problem solving.
  • 6.1 The student represents and uses rational numbers in a variety of equivalent forms. The student is expected to: (B) generate equivalent forms of rational numbers including whole numbers, fractions, and decimals
  • The student solves problems involving direct proportional relationships. The student is expected to: (B) represent percents with concrete models, fractions, and decimals (7.1) The student represents and uses numbers in a variety of equivalent forms. The student is expected to:  (B) convert between fractions, decimals, whole numbers, and percents
  • CCSS 6.NS.4 –  GCF, LCM
  • CCSS..5.NF.A.1 - +/- fractions with unlike denominators
  • CCSS.5.NF.A.2    Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators
  • CCSS..5.NF.B.3 Interpret a fraction as division of the numerator by the denominator
  • CCSS 5.NF.4a-1, 4a-2, 4b-1,6-1,7a,7b,7c ; 6.NS,1-2- Multiplying and dividing with fractions
  • CCSS 5.NF.3-1, 3-2 Interpreting fractions


Learners will be able to....

  • Apply their mathematical knowledge, skill and reasoning, specifically that of operational and rational numbers (fractions, decimals & percents), and problem solving strategies to determine and justify answers to authentic problems
  • Apply their mathematical knowledge, specifically that of operational and rational number to their everyday lives



Learners will understand that...

  • Rational numbers can be represented as decimals, fractions, and percents
  • Strategies can be used to simplify expressions and to compare and order rational numbers
  • There are advantages and disadvantages to each type of representation (fractions, decimals, and percents)
  • Sometimes the “correct” unit rate (i.e., fractions, decimals, and percents) is not the best solution to real world problems
  • To compare fractions and decimals the numbers must be converted to the same form

Essential Questions

Learners will keep considering...

  • Why do we need rational numbers?
  • How are rational numbers used in our everyday lives?
  • How are mathematical ideas interconnected and build on one another to produce a coherent whole
  • Why do we need to compare rational numbers and which form is best to use given a specific situation?


Learners  will know…

  • Key Term definitions – unit rate, percent, decimal, mixed & improper fractions, rational number  
  • The properties of rational numbers expressed in a variety of forms
  •  Strategies to convert between rational numbers.  
  •  When comparing fractions, a common denominator is essential
  • There are key questions to ask when solving a problem

Learners will be skilled at...

  • Computing with rational numbers being expressed in a variety of forms  
  • Determine if a solution is appropriate and moving beyond a particular problem by thinking of other situations
  • Computing the LCM, GCF to help solve problems
  • Simplifying fractions.  
  • Generating equivalent forms of rational numbers.  
  • Converting between fractions, decimals, whole numbers
  • Calculating rational numbers using the 4 operations
  • Identifying and calculating exponents and powers of numbers
  • Using various problem solving strategies to solve authentic problems
  • Asking themselves key questions when solving problems

Stage 2- Evidence


Learners will show that they understand by evidence of…

Set up a Sale Item  (from Yummymath.com)

GRASPS Task written in student-friendly language:


Your goal is to purchase the ‘in’ hoodie, by convincing your guardian to buy it for you today because it is only on sale for one day, at J Crew.


You want the hoodie to fit in with your peers.  You know your guardian always likes the idea of a bargain as s/he is thrifty. Therefore you have to convince your guardian that s/he is saving x amount of money by buying it today, as the sale price includes up to 3 discounts.  


Your mother /father / guardian, as they have the money needed to purchase the hoodie.


J Crew have this particular hoodie on sale for one day only.  You therefore need to prove to your guardian that by buying today, they could save x amount of money.  You know that your guardian likes to think about things, therefore you need to record your thinking, so they can go back to your explanation multiple times before making their final decision.


You are to show your guardian how much money they are saving by purchasing the hoodie today.  Not only will it include an oral explanation, but you need to record the savings they will make in some form. For example, a spreadsheet, a tutorial demonstrating how all the discounts affect the price  - it is up to you.


Learners  will show they have achieved Stage 1 goals by demonstrating their understanding through the Six Facets of Understanding...


Their reasoning behind their solutions to every problem based lesson


What information they have been given, and what is needed to solve the problem


What operational / mathematical skills they need to solve the problem

Have perspective

To visualize how the problem could be solved, and how it can be solved in different ways, depending on the perspective that it has been interpreted. Therefore considering various possibilities, choose the best method to reach the determined goal.


Be able to put themselves into the situation to be able to understand various perspectives, therefore enabling them to choose the best solution possible for the given problem so the most people involved are satisfied with the outcome.

Have self-knowledge

Be able to build on their prior knowledge, and be aware of their best learning style eg. manipulatives, paper to solve the specific problem, taking time to reflect and consider alternative options if the answer doesn’t seem correct.  Know it is ok to make mistakes to move forward.

Stage 3- Learning Plan

Time Frame

Learning Events

Progress Monitoring


As this is a very individualized class,  the students work at their own rate, at their level, on whichever skill they need to develop (identified from the pre-test), therefore time is dependent on their mastery and depth of understanding.

Likewise, how the student wishes to practice their skill is dependent upon the way they prefer to learn, therefore they are given a choice of paper / on-line games or hands on manipulatives/ partner games to choose from to reinforce their current learning

As this is individualized, within the lesson time frame, a student will have one on one with a teacher, which may include a mini lesson, on-going formative feedback, and practice - this may be individually or with a partner or small group.

Formative feedback will be given daily

Each student will be involved in demonstrating the transfer of their new skills / knowledge to  solve an authentic problem using problem based learning and the problem solving framework  - this may be individually / partner / group, depending on the situation

Resources / Materials:

  • Link to teacher resource of menu created for student options to reinforce their learning of a specific skill.  This is modified, then shared with students via google doc.