Big Idea(s): Fractions

Expectations:

Number

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

  • B1.7 recognize that one half and two fourths of the same whole are equal, in fair-sharing contexts

  • B1.8 use drawings to compare and order unit fractions representing the individual portions that result when a whole is shared by different numbers of sharers, up to a maximum of 10

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

  • B2.5 represent and solve equal-group problems where the total number of items is no more than 10, including problems in which each group is a half, using tools and drawings

Social Emotional Learning Skills 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 of representing (select from and create a variety of representations of mathematical ideas (e.g., representations involving physical models, pictures, numbers, variables, graphs), and apply them to solve problems) and 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) so they can make connections between math and everyday contexts to help them make informed judgements and decisions

Watch this first video #1 about what polar bears eat at the zoos. Watch the second video #2 about rectangular frozen polar bear treats with the students. (It is not necessary to watch them in their entirety.)

Ask students these questions:

• What do you see in this video?

• Have any of you ever been to the zoo?

• What do you know about polar bears?

• What do polar bears eat at zoos?

• Describe the daily diet of polar bears.

• What do polar bears like to eat as treats?

Role-playing

Imagine that we get to go to the zoo for a day and work with the zookeepers as their assistants. One of our jobs would be to prepare food for the polar bears and to make special frozen blocks of treats for 3 different groups of bears and to feed them.

Day 1

Present slides #2, #3, and #4 to the students -- a photo of 2 polar bears, polar bear diet, and polar bear food boxes.

The zoo veterinarian has asked us to make two daily food boxes, one each, for the 2 polar bears at the zoo. The daily meal boxes are in the shape of rectangular prisms for 2 polar bears.

Present slides #4, #5, #6, #7, #8, #9, #10, and #11 - outlines of the different foods that make up the bears’ diet.

Explain that the vet has given us the following sets of food to divide between our 2 polar bears:

• 2 pieces of meat

• 4 heads of lettuce

• 5 herring fish

• 7 carrots

• 8 melons

• 9 apples

• 10 sweet potatoes

Tell the students it is their job to divide the above food equally between the 2 polar bears so that each bear receives the same amount of each type of food. Draw their attention to slide #4.

Pose the question, “How will we solve this problem?”

Have the students turn to another student, and use the strategy of “Think-Pair-Share” partner discussion for brainstorming. Hold a large group discussion.

Instruct the students to solve the problem. Organize them into groups of 2 or more.

Attach the chart paper to the wall (classroom, hallway, etc.) Have the child work vertically. (Research suggests that working on vertical surfaces improves learning by bringing the task closer to the children’s eyes.)

Students use the methods, strategies and manipulatives of their choice to solve the problem.

Students record their mathematical reasoning used to solve the problem on the chart paper. Students are instructed to write and print in marker so they cannot erase their work. Teachers need to see all traces of their thinking, even errors and corrections made.

Once students have demonstrated how the sets of food items are divided fairly between the 2 polar bears, determine whether the students understand that each polar bear has received half of the total food. Post the question, “What fraction of the food did each polar bear receive?”

Day 2

Re-watch the second video #2 about retangular frozen polar bear treats with the students.

Tell the students that the veterinarian has now asked us to make three special frozen fruit and vegetable loaves for each set of bears in the zoo to share.

Show Images #12, #13, and #14 to the students - the 3 sets of bears i.e., 2 polar bears, 4 grizzly bears, and 10 black bears. Tell them that these are all the sets of bears in the zoo. Each group or set lives together in one enclosure.

Explain that each loaf is slightly different for each group of bears, because the 3 types have slightly different diets.

Show students the image #15, the frozen treat maker container. This is one of the boxes that we fill up with treats and water, and then freeze.

Explain that we only have three large, plastic rectangular bins big enough to make three frozen treats at a time. What does that mean for each bear group? (It means that we can only put one loaf in each of the 3 bear enclosures.)

If we can only put one loaf in each bear enclosure, what does that mean for the bears? (It means the bears in each enclosure will have to share until we can make more frozen treats.)

Draw students’ attention back to images #12, #13, and #14. What will we have to do to each frozen ice block treat? (There is a different number of bears in each of the 3 enclosures, so we will have to divide each of the three frozen rectangular blocks of ice into a different number of pieces:

• 2 polar bears

• 4 grizzly bears, and

• 10 black bears.)

Pose the question, “For each set of bears, how should we divide each frozen loaf up to fair sharely within each group?”

Instruct students to show how much of the rectangular prism-shaped treat each bear will get in the following situations:

• each polar bear if 2 polar bears share one frozen loaf

• each grizzly bear if all 4 of them share one frozen loaf.

• each black bear if all 10 black bears share one frozen treat.

Organize the students to work in pairs or small groups to solve the problem.

Attach enough chart paper for each group to the wall (classroom, hallway, etc.). Have the children work vertically. (Research suggests that working on vertical surfaces improves learning by bringing the task closer to the children’s eyes.)

Students use the methods, strategies and manipulatives of their choice to resolve the problem.

• Students record the mathematical reasoning they used to solve the problem. Students are instructed to write and print in marker. We want to see all traces of their thinking, even errors and corrections made.

Bansho: (This strategy is recommended in the middle and intermediate grades. However, it is easily achievable in the primary grades).

The teacher encourages a whole class discussion in which students explain the mathematical reasoning used in their solutions, methods and strategies.

First, before the official Bansho begins, pair two teams together. Each team presents to the other. This gives every team a chance to talk about their work. While they do that, the teacher circulates and listens for evidence of understanding. This helps the teacher select the teams that will present during the congress. Furthermore, when presenting to the other team, students sometimes have an “aha!” moment and/or notice and correct misunderstandings.

Next, post the chart paper of the 2 or 3 groups chosen by the teacher on the board. Explain the Bansho to the students :

• The 2 or 3 selected groups present their work and the strategies used, one group at a time.

During the presentations, the other students listen to and analyze their peers' mathematical reasoning. Students are invited to ask mathematical questions to the other groups.

Ask the students who are listening if they had the same reasoning or a different reasoning. This will make them compare their work with the one presented and engage them even more.

Students could upload their work to Jamboard or to a Google Slide deck. Notes and comments could be added as the presentation is being made.

The teacher asks open-ended questions to further their thinking. (The Art of Effective Questioning)

First Problem:

• What questions did you ask yourself while working?

• How did you feel during work?

• Why did you make this decision or choose this strategy to solve the problem?

• What changes did you make to solve the problem?

• What was the most difficult part of the task?

• What strategy did you use? Did your strategy work?

• What are the steps that you took to solve the problem?

• How did you divide the food (2 pieces of meat, 4 heads of lettuce, 5 herring fish, 7 carrots, 8 melons, 9 apples, and 10 sweet potatoes) between the polar bears? Are there any remaining pieces? Possible answer: We gave 1 piece of meat, 2 heads of lettuce, 2 herrings, 3 carrots, 4 melons, 4 apples, and 5 sweet potatoes to each animal. We had 1 herring, 1 carrot, 1 apple left so we cut them into 2 pieces (halves) and gave a piece (a half) to each animal.

• What is different about dividing a set of 8 melons between 2 polar bears and dividing one melon between two polar bears? What is the same?

• How can 4 of 8 melons be half but one half of one melon be one half? Are they the same? What is different about them?

Second Problem:

• How did you divide one frozen treat loaf between the 2 polar bears? Are there any remaining pieces? Possible answer: We cut the loaf into halves, which gave us a total of 2 pieces, so each animal received one half of the loaf. No remaining pieces.

• How did you divide the one frozen treat loaf among the 4 grizzly bears? Are there any remaining pieces? Possible answer: We cut the loaf into 4 pieces. They each got one quarter of the loaf. No remaining pieces.

• How did you divide the one frozen loaf among the 10 black bears? Possible answer: We cut the loaf into 10 pieces. They each got one tenth of the loaf. No remaining pieces.

• What model or manipulative did you use to solve the problem?

• What fractions of snacks are each group of bears receiving? What do you notice about the size of the portions? Possible answer: The more animals there are in the group, the smaller the portion each group member gets.

• Which group of bears got bigger pieces of the frozen treat loaves? How do you know? Why does this happen?

After the presentations, invite the other groups to place their solutions below the posted one that most closely resembles their own. At this stage, students learn to discern similarities and differences in mathematical reasoning, methods and strategies. If the group has used a different strategy, they can place it separately on the board.

Opportunities for Assessment

Assessment for Learning

• Conversations: Ask students questions to check their understanding of the problem. What strategies did they use?

Evaluation as learning

• Observations: Observe the students and their ability to explain the reasoning behind the choices they make. Observe students and see how they express themselves and organize themselves.

• Conversations: Observe students and see how they express themselves and organize themselves in teamwork. Listen to conversations between students. Encourage class and small group conversations that allow students to clearly express their thoughts and develop their thinking.

SEL Self-Assessments (English) and Teacher Rubric

Provide students with an exit ticket such as the one below. This activity will further consolidate understanding of equal-group problems where the total number of items is no more than 10, problems in which each group is a half, and problems that highlight the importance of the size of the whole in governing the number of equal groups in the12. The size of the fraction12is relative to the whole. So12of a carton of 6 eggs compared to12of a carton of 10 eggs is different. We should ask the students, even though Martin and Lucy each gave12of their egg carton, why did the animals get different quantities?

Monkeys and apes love raw eggs. Martin had a carton of 6 eggs. He gave12of his eggs to the monkey. Lucy had a carton of 10 eggs. She gave12of her eggs to a baboon. Did the monkey and the baboon get the same amount of eggs? How do you know? Which animal got the most food? How many eggs did each animal get?