Big Idea(s):

Fractions as equal shares

Expectations:

Number

• B1. demonstrate an understanding of numbers and make connections to the way numbers are used in everyday life

  • B1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths

• B2. use knowledge of numbers and operations to solve mathematical problems encountered in everyday life

  • B2.7 represent and solve problems involving multiplication and division, including problems that involve groups of one half, one fourth, and one third, using tools and drawings

Social Emotional Learning Skills (SEL) in Mathematics and the Mathematical Processes

A1. Throughout this grade, in order to promote a positive identity as a math learner, to foster well-being and the ability to learn, build resilience, and thrive, students will apply, to the best of their ability, a variety of social-emotional learning skills to support their use of the mathematical processes and their learning in connection with the expectations in the other five strands of the mathematics curriculum.

In this lesson, to the best of their ability, students will learn to think critically and creatively as they apply the mathematical processes reasoning and proving (develop and apply reasoning skills (e.g., classification, recognition of relationships, use of counter-examples) to justify thinking, make and investigate conjectures, and construct and defend arguments), selecting tools and strategies (select and use a variety of concrete, visual, and electronic learning tools and appropriate strategies to investigate mathematical ideas and to solve problems), and communicating (express and understand mathematical thinking, and engage in mathematical arguments using everyday language, language resources as necessary, appropriate mathematical terminology, a variety of representations and mathematical conventions), so they can make connections between math and everyday contexts to help them make informed judgements and decisions.

DAY 1:

Opportunities for Differentiation

Parallel tasks:

• For students who would benefit from more experience working with halves, provide the following problem: two dogs share five pieces of jerky.

• For students who would benefit from a more challenging experience provide the following problem: five dogs share 14 pieces of jerky.

• Offer students paper strips that may be used as a tool.

• If students who are solving the six dogs, ten strips of jerky scenario tend to focus on making halves, ask them to consider other sharing quantities e.g., “ You made two (or four) pieces. What other ways could you divide the strip?”

Opportunities for Assessment

Ask students to explain the strategies they are using. How does this strategy help ? What makes it hard to use ?

Note their ability to explain the reasoning behind the strategies they try.

Make anecdotal notes of actions such as:

• Verbalizes the problem

• Tries multiple strategies

• Checks the reasonableness of the solution

• Use fractional language

• Communicates ideas orally, gesturally, and in written form

• Records thinking so others can understand the solution.

Students online may wish to use slides from Sharing Doggie Treats to help them solve the problems and communicate their thinking.

DAY 2:

Opportunities for Differentiation

• Let students discuss the exit ticket with a partner before completing the task.

• For students who completed a parallel task, review the premise of the exit ticket, that is, 10 strips each cut into six pieces shared between six dogs.

Parallel tasks:

• For students who worked with the half problem on day one provide the following problem to work with thirds: three dogs share four pieces of jerky.

• For students who would benefit from a more challenging problem provide the following problem: seven dogs share five pieces of jerky. Be sure they use a visual as well as numeric representation to explain their thinking.

Opportunities for Assessment

Ask students to explain the strategies they are using. “How does this strategy help ? What makes it hard to use ?” Note their ability to explain the reasoning behind the strategies they try.

Make anecdotal notes of actions such as:

• Verbalizes the problem

• Tries multiple strategies

• Checks the reasonableness of the solutions

• Can describe the solution

• Records thinking so that others can interpret the information

• Uses fractional language to explain thinking

Be aware of students' ability to explain the reasoning behind the strategies they tried.

Use the Bansho technique to display all the solutions the students developed, organized by strategies used. Students have the opportunity to see and hear many approaches. Help them realize that the same strategies can be used with different starting numbers and with improper and proper fractions. What strategies are similar? How are they different ?

• Do you see a method that makes sense to you even if it is different from your own ?

• Which strategy would you use if you solved another problem like this again ?

• How did this strategy help you ? What made it difficult to use ?

• Do you see a method that is different from your own that you like better ? Why do you like it better ?

• How do you know this solution is accurate ?

• How can we tell that different-looking answers are equally correct ?

• How could you prove that this answer is right ?

• What fraction of a strip is one small piece? How do you know ? (If you have parallel tasks displayed ask: Why is there a different fractional value for this problem ?)

Compare a solution from day 1 and one from day 2 that use the same strategy (e.g., cut all the strips into small pieces before doing a one to one fair share). Explore the similarities and differences between the fractional amounts and how they are represented and described.

Assessments as, for and of learning are embedded throughout the lesson.

SEL Self-Assessments (English) and Teacher Rubric

At another time, discuss with the students:

• “Why must all pieces be the same size?”

• “When might Adrianna decide to share the jerky unfairly?”

• “When is fair sharing important in your life?”

• “When might we get unequal shares?”

Encourage them to recognize the links between the math task and real life. How does thinking about imaginary dog jerky help them consider fairness in the class, the family, the community, or the wider world ?

Have the class brainstorm ideas together based on the sentence stem “I have to share when….” Then have them reflect on which sharing examples require equal sharing. Then discuss how the sharing makes them feel ? How do they cope when they don’t want to share but are required to do so ? Is sharing always important? When does it really matter ? When is it ok not to share ?

Have the students try a further doggie day care task - this time Adrianna is sharing 12 dog biscuits amongst four dogs. What fraction of the biscuits does each dog get ? What if she has 13 biscuits or 14 biscuits ?

Read aloud stories based on fair sharing. As you read the book, think aloud using fractional language, noting the importance of equal units, etc. Two possible stories available in print and online are The Doorbell Rang by Pat Hutchins or The Lion’s Share by Matthew McElligott, but there are many others.