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This activity is created for second grade students and focuses on geometric reasoning. In this activity, students will identify the perimeter of polygons with whole number side lengths.

Students will be placed in heterogenous groups of four and will be provided a bucket of a variety lego pieces. Each circle on the lego piece is counted as 1 unit. For instance, a lego piece with two groups of four circles would have an overall perimeter of 12. There will be stations around the room with each station containing an instruction card. Groups will rotate around the room, read the instruction card which provides the students with a perimeter limit and a polygon that must be created with these limits, and begin to work. When each student has finished they will draw a picture of their polygon and perimeter then move onto the next station. The main objective is for students to perform a variety of tasks related to perimeter. The sociocultural theory is being used by having students work in groups of four to solve each perimeter task card. While the constructivist theory is being used by having students have a choice in their building process, building design, and placing students of different abilities together to help one another. Allowing students to take part in constructing perimeter through building, students are using real life experience which could allow them to find the perimeter of houses, their room, and much more. From this activity, students will have the knowledge to apply perimeter to real life objects rather than from pictures on a paper. Students are engaging in discussions with their groups to solve the task card and illustrate their findings. They are working in groups to expand their knowledge.

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To access prior knowledge, encourage inquiry, and pose a problem, the teacher will act as if she is having difficulty solving the problem 3+8x2 since she is coming up with a different answer than the calculator. Using what they know about addition, subtraction, and working from left to right on basic equations, most students will answer 22. When the answer on the calculator is shown, they will see that it is different. Students will wonder why the answer is different and work in groups using counters or pictorial representations to try to answer why the answer is different. Students will share their thoughts and compare their thinking with other students. The answer is then revealed and explained that 8x2 must be solved first for the answer to be 19. Ask students if knowing what to do first is important? Have them explain their answer. Since knowing what to do first is not obvious when observing the problem, ask students what could be included to the problem to make known what must be solved first. Have students do a think-pair-share to discuss the question.

Students will analyze two problems using the same numbers but with the parentheses in a different place (ex: ( 5+3)x2=? and 5+(3x2)=? ) Model how to solve each problem using think aloud to see if the placement of the parentheses make a difference. Ask how the problems and answers are similar/different. Give students similar problems to see how the placement of the parentheses makes a difference in the answer. Have students work in groups. Discuss after each problem is solved. They will realize that the placement of the parentheses and the order in which the problems are solved matters. Explain that not all equations have parentheses like the first problem they tried to solve. Introduce to order of operations rule. Demonstrate examples of how to apply the rule to problems without parentheses. Have students  work on a few problems together and review as they go along. Gradually increase complexity.

The socialcultural theory is represented in this activity as students learn from each other while they work and communicate together in small groups and pairs to solve problems and address inquiry.  The constructivist theory is also represented as prior knowledge is accessed, inquiry is encouraged, and students explore, analyze, and discuss questions posed and determine the best way to solve problems. The lesson is designed to scaffold knowledge step-by-step through guided instruction, modeling, and providing examples as students construct their own knowledge building on what they already know.

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·      In the sociocultural theory, Vygotsky states that you cant see or measure a students development individually; you have to also see the students social world where they developed.

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In the Social cultural theory, it defines that skills

that can be developed with adult

guidance or peers around exceeds

what can be attained alone.

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"Through participation in activities that require cognitive and communicative functions,
children are drawn into the use of these functions in ways that nurture and 'scaffold' them"
The main idea of the theory is that the ways people interact with others and the culture they live in shape their mental abilities

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One major concept in social cultural theory is Semiotic Mediation.This is referring to use of language and different way to show cultural practices such as diagrams, pictures, and visuals.

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The nature of all learner is affected by not just the culture the teacher creates, but of the social and cultural aspects of the members in the classroom.

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Constructivist theory is a theory created by Jean Piaget and its an approach that people actively construct or create their own knowledge and that reality is determined by the experiences of the learner.

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Two ways this can happen are by assimilation and accommodation.
Assimilation is when you have a new concept and it fits or goes with something you already know.
Accommodation is when the new concept does not match or fit with anything prior, therefore the brain revamps or replaces it.

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Generally prior knowledge is good, but sometimes misconceptions and wrong information can be a hindrance. Sometimes
time must be spent correcting prior knowledge before new learning can occur.

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In constructivist theory, it is important that the teacher lets the student construct or create questions, theories and answers during the lesson. This way the teacher can see where the student stands.

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In this activity, which has been designed for a third grade classroom, the students will be able to write fractions in mathematical notations by questioning, summarizing and clarifying through the use of manipulatives.

The students will be divided in small heterogeneous groups of 4-5 students each. They will be provided a dry erase board as well as a pack of candy with a paper plate. The students will then be asked to open their packs over the plate and count each candy piece that comes in their individual packs. I will then ask the students to place all of the candy pieces on the bottom of the fraction and ask them to think and tell me what that bottom number is called and why. Upon listening to the different answers, I will continue by explaining that the bottom number is the denominator because it represents the whole. I will ask the students to write this number on their dry erase board as the denominator. I will continue by asking the students to focus on just the green pieces of candy and count how many they have. The students will then be asked to move all of the green pieces of candy to the top of their plate. I will ask them to explain why these pieces go at the top of the plate. Upon listening to their reasoning, I will clarify that this is the numerator because it is how many parts of the whole we are counting. The students will then be asked to write that number as the numerator and then read the fraction as they have written it on their dry-erase board. I will repeat this process with different colored candy pieces to give the students practice by creating their own different fractions with the candy pieces.

At the end of the activity, I would ask for a volunteer to identify the numerator and denominator in the fraction written on the board and ask the student to create a matching visual that represents the fraction. I will also ask the students discussion questions such as: What do you notice about the denominators in all of these fractions? How does that fraction look compared to the others?  What do you notice about the numerators?

Through this activity, the students will expand their knowledge on how to create fractions with items that they may be able to find in their everyday life, such as candy. The purpose of this activity is to allow the student to reach their optimal learning zone by scaffolding, sharing and comparing their findings with their peers.