Teaching and Assessing Junior High School Mathematics (Introductory)

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COMPLETE HANDOUT

 

NUMBER SYSTEM

UNIT 1 

SCHOOL MATHEMATICS CURRICULUM STANDARDS-BASED VERSUS OBJECTIVE-BASED CURRICULUM

 

STANDARDS FOR LEARNING AND TEACHING MATHEMATICS

Educational decisions made by teachers, school administrators, and other professionals have important consequences for students and for society. The Principles for school mathematics provide guidance in making these decisions.

The five Process Standards are described through examples that demonstrate what each standard looks like and what the teacher's role is in achieving it:

· Problem Solving

· Reasoning & Proof

· Communication

· Connections

· Representation

Problem Solving

Problem solving in math means “becoming involved in a task for which the solution method is not known in advance. To find a solution, students must use previously acquired knowledge and, through this process, gain new mathematical understandings” (Bahr & Garcia, 2010). Problem solving should be an integral part of daily math. Problems can be drawn from real life experiences and applications. The problems selected should be carefully analyzed so the teacher can predict the mathematical ideas that will underscore students’ problem solving efforts.

To promote problem solving, students must feel safe in their learning environment. They need to know they are free to explore, take risks, share, and argue strategies with one another. This type of environment will also allow them to confidently work through problems and challenges, and view themselves as capable mathematical thinkers. Specific problem solving strategies, or heuristics, is also a focus of the teacher. This includes things like simplifying the complexity of a problem, drawing a diagram or picture of a problem, seeing patterns, guessing and checking, and working backwards.

Problem solving also has a metacognitive component. Teachers can promote this level of thoughtfulness by asking questions:

· What do we know from the information in this problem?

· What information does it ask for?

· Is there any missing information that might make solving the problem easier?

· What should we do first, second, etc.?

· Are we getting closer to the solution?

· Should we try something else?

· Why is that true?

These types of questions place the responsibility for success on the students.

Communication

Mathematics communication is both a means of transmission and a component of what it means to “do” mathematics. Teacher have to provide an environment in which students can risk expressing their beginning efforts to communicate their thinking. Teachers must be patient while students begin to do this, because communicating in math doesn’t come naturally to students.

The NCTM standards provide a complete list of standards in communication and the benefits of communicating in math.

When students share, they should genuinely listen to one another, compare it to their own ideas, evaluate it, then share their own opinions. Teachers can use probing and prompting questions during discussions as scaffolding. In older grades, students should be encouraged to elaborate more.

Writing in math is also beneficial to deepening mathematical understanding. In the primary grades, they rely more on pictures and as they get older will be able to form more complete sentences and thoughts. Writing in math also allows them to practice using mathematical vocabulary and symbols. Their writing skills are consequently enhanced as they practice justifying and writing in this expository form. Just like writing in any other content area, the teacher will have to model how this should be done effectively.

Reasoning and Proof

Reasoning is a habit and should provide a context for developing important mathematical ideas. Questioning is the key! Ask WHY? Mathematics involves discovery, so invite students to make conjectures and create, refine, and evaluate them. Also, allow students to explore and explain their own reasoning. It’s often best to start students off with what they know, and then build from there.

This is where students can take advantage of manipulatives and using technology to solve problems and explore their conjectures. Several virtual manipulative websites exist to get the same practice with manipulatives while utilizing technology, especially if the manipulatives in the classroom are limited. One such website is found here. Using manipulatives will help reinforce concepts to all students, especially students with learning disabilities and English language learners.

Sometimes younger children need a discrepant or contradictory event to verify their reasoning. They tend to overgeneralize an idea, which means they may apply reasoning from one context to a context where the same reasoning does not really apply. Through enough exploration and discovery the students will be able to accommodate and assimilate the new mathematical reasoning into the correct schema.

Also encourage students to look for patterns. These patterns can be spatial, temporal, logical, and sequential.

Representation

This is asking students to show a mathematical idea in more than one way. There are five ways to represent thinking:

1) Manipulative models
2) Static pictures
3) Written symbols
4) Spoken/written language
5) Real-world situations or contexts

 

Real-life situations are very valuable to the students because it gives them something more concrete to work with, and they begin to see the real purpose and meaning behind using the mathematical concept. Because adults think in symbols and children do not, children support their thinking with examples they have seen in the real world.

Representations are used by children first to display the problem, then to find a solution, and finally use tools to solve similar problems. This will especially be useful to special needs children and English language learners to use situations they are familiar with.

Connections

Connecting is the experience of mentally relating one object to another” (Bahr & Garcia, 2010). Elementary Mathematics is Anything but Elementary (2010) identifies six types of connections distinguished by what types of thoughts are being connected:

1) Representations

2) Problem solving strategies or conjectures

3) Prior and current math learning

4) Mathematical topics

5) Mathematics and other subjects

6) Mathematics and real-life situations

If you encourage these connections, it will increase your students’ mathematical reasoning abilities. One of the roles of teachers is to compare strategies students share and help students see the connections between those strategies. It is essential to make these connections with prior math learning, so the learning is logical and builds from what students know. Just when you build a structure, you need to have a firm foundation before you can start building the structure. The same is true in math. When students make connections between math concepts it’s like they have formed a neighborhood of strong buildings and can see how the neighborhood works and functions together, instead of each building functioning or existing on its own. Children learn about the world in connected ways, so balanced math instruction will help children do this.

Integrating other subjects with math has to be meaningful. Therefore, if you simply read a book with a math concept in it, that is not successful integration.  “When mathematics is consistently used to solve problems in other subject area contexts, connections, to real life occur consistently” (Bahr & Garcia, 2010).

 

THE SIX PRINCIPLES ADDRESS OVERARCHING THEMES:

· Equity. Excellence in mathematics education requires equity—high expectations and strong support for all students.

· Curriculum. A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well-articulated across the grades

· Teaching. Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

· Learning. Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

· Assessment. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.

· Technology. Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.

The Standards for school mathematics describe the mathematical understanding, knowledge, and skills that students should acquire from prekindergarten through grade 12. Each Standard consists of two to four specific goals that apply across all the grades.

THE FIVE CONTENT STANDARDS EACH  ENCOMPASS SPECIFIC EXPECTATIONS, ORGANIZED BY GRADE BANDS:

· Number & Operations

· Algebra

· Geometry

· Measurement

· Data Analysis & Probability

AIMS OF MATHEMATICS EDUCATION IN BASIC SCHOOL (UP TO JHS)

The mathematics curriculum is designed to help learners to:

1. Recognize that mathematics permeates the world around us

2. Appreciate the usefulness, power and beauty of mathematics

3. Enjoy mathematics and develop patience and persistence when solving problems

4. Understand and be able to use the language, symbols and notation of mathematics

5. Develop mathematical curiosity and use inductive and deductive reasoning when solving problems

6. Become confident in using mathematics to analyse and solve problems both in school and in real-life situations

7. Develop the knowledge, skills and attitudes necessary to pursue further studies in mathematics

8. Develop abstract, logical and critical thinking and the ability to reflect critically upon their work and the work of others.

STANDARDS-BASED CURRICULUM

The standards-based curriculum or the intended curriculum is the official or adopted curriculum contained in state or district policy. A body of content knowledge that students are expected to learn based on their participation within the school experience, standards-based curriculum includes broad descriptions of content areas and often specifies performance standards that students are expected to meet. State and district assessments are linked directly to the content and performance standards contained in the standards-based curriculum. The standards-based curriculum outlines graduation requirements, which are taken from state department of education guidelines that specify the subjects and skills that should be taught at each grade level.

A standards-based curriculum refers explicitly to specific knowledge, learning experiences to gain that knowledge, and assessments to check for mastery of that knowledge, developed by looking at the standards of a district, state, or nation.

Standards-based curriculum helps teachers to link the taught curriculum to the required standards. It is the connection between the content standards and the taught and learned curriculum.

Taught Curriculum

The daily events that occur in the school community, including all lessons, activities, and social gatherings among peers. The techniques used by teachers during instruction, such as lectures and discussions, are part of the taught curriculum. The term even refers to the style of instruction used by the teacher (e.g., group arrangements, time allocated for instruction, teachers’ personal beliefs and attitudes related to the intended curriculum). Curriculum materials strongly influence the instruction that occurs in classrooms, but the use of curriculum materials varies considerably. In other words, teachers determine the variety of activities and lessons that get taught and how their students will be asked to use the information they receive. Textbooks, worksheets, and electronic media are all examples of curriculum materials that are part of the taught curriculum. However, it should be noted that teachers often mistakenly refer to the school district-adopted textbooks as “the curriculum.”

Learned Curriculum

The information that students learn as a result of being in the classroom and interacting with the taught curriculum is the learned curriculum. This can include information that may not be a part of the standards-based or taught curriculum, something that can be problematic when the learned curriculum includes erroneous or incomplete information. It is important to use assessments that accurately indicate how much of the standards-based curriculum that students have actually achieved.

In order to become familiar with the standards, Ms. Begay must find answers to those questions:

· What domains or areas will be covered?

· What knowledge, competencies, and skills will be attended to?

· What type of learning skills will be required?

She must also see the scope and sequence of the standards and how they expand across the grades. What instructional teaching skills and techniques are necessary for her to effectively teach the required content to her students? She can’t just teach to one grade level. She must be aware of the content her students were taught and expected to learn last year, and she needs to know what her students will be taught in future grades so that she can teach them accordingly. For further illustration of this idea, review the timeline below.

OBJECTIVE-BASED CURRICULUM

Objectives are statements that describe the end-points or desired outcomes of the curriculum, a unit, a lesson plan, or learning activity. They specify and describe curriculum outcomes in more specific terms than goals or aims do. Objectives are also the instructions or directions about what educators want the students to be able to do as a result of instruction. Considered essential to goal setting and planning curricula, objectives aid students, teachers, and parents by specifying the direction of the curriculum and goals. Typically written by school districts, schools, and individuals, objectives also help ensure that educational processes are aligned and that instructional activities are directed toward the defined outcomes or learning.

BENEFITS OF OBJECTIVE-BASED CURRICULUM

Clarity

The focus on outcomes creates a clear expectation of what needs to be accomplished by the end of the course. Students will understand what is expected of them and teachers will know what they need to teach during the course. Clarity is important over years of schooling and when team teaching is involved. Each team member, or year in school, will have a clear understanding of what needs to be accomplished in each class, or at each level, allowing students to progress. Those designing and planning the curriculum are expected to work backwards once an outcome has been decided upon; they must determine what knowledge and skills will be required to reach the outcome.

Flexibility

With a clear sense of what needs to be accomplished, instructors will be able to structure their lessons around the student’s needs. OBE does not specify a specific method of instruction, leaving instructors free to teach their students using any method. Instructors will also be able to recognize diversity among students by using various teaching and assessment techniques during their class. OBE is meant to be a student-centered learning model. Teachers are meant to guide and help the students understand the material in any way necessary, study guides, and group work are some of the methods instructors can use to facilitate students learning.

Comparison

OBE can be compared across different institutions. On an individual level, institutions can look at what outcomes a student has achieved to decide what level the student would be at within a new institution. On an institutional level, institutions can compare themselves, by checking to see what outcomes they have in common, and find places where they may need improvement, based on the achievement of outcomes at other institutions. The ability to compare easily across institutions allows students to move between institutions with relative ease. The institutions can compare outcomes to determine what credits to award the student. The clearly articulated outcomes should allow institutions to assess the student’s achievements rapidly, leading to increased movement of students. These outcomes also work for school to work transitions. A potential employer can look at records of the potential employee to determine what outcomes they have achieved. They can then determine if the potential employee has the skills necessary for the job.

Involvement

Student involvement in the classroom is a key part of OBE. Students are expected to do their own learning, so that they gain a full understanding of the material. Increased student involvement allows students to feel responsible for their own learning, and they should learn more through this individual learning. Other aspects of involvement are parental and community, through developing curriculum, or making changes to it. OBE outcomes are meant to be decided upon within a school system, or at a local level. Parents and community members are asked to give input in order to uphold the standards of education within a community and to ensure that students will be prepared for life after school.

DRAWBACKS OF OBJECTIVE-BASED CURRICULUM

Definition

The definitions of the outcomes decided upon are subject to interpretation by those implementing them. Across different programs or even different instructors outcomes could be interpreted differently, leading to a difference in education, even though the same outcomes were said to be achieved. By outlining specific outcomes, a holistic approach to learning is lost. Learning can find itself reduced to something that is specific, measurable, and observable. As a result, outcomes are not yet widely recognized as a valid way of conceptualizing what learning is about.

Assessment problems

When determining if an outcome has been achieved, assessments may become too mechanical, looking only to see if the student has acquired the knowledge. The ability to use and apply the knowledge in different ways may not be the focus of the assessment. The focus on determining if the outcome has been achieved leads to a loss of understanding and learning for students, who may never be shown how to use the knowledge they have gained. Instructors are faced with a challenge: they must learn to manage an environment that can become fundamentally different from what they are accustomed to. In regards to giving assessments, they must be willing to put in the time required to create a valid, reliable assessment that ideally would allow students to demonstrate their understanding of the information, while remaining objective.

Generality

Education outcomes can lead to a constrained nature of teaching and assessment. Assessing liberal outcomes such as creativity, respect for self and others, responsibility, and self-sufficiency, can become problematic. There is not a measurable, observable, or specific way to determine if a student has achieved these outcomes. Due to the nature of specific outcomes, OBE may actually work against its ideals of serving and creating individuals that have achieved many outcomes.

Involvement

Parental involvement, as discussed in the benefits section can also be a drawback, if parents and community members are not willing to express their opinions on the quality of the education system, the system may not see a need for improvement, and not change to meet student’s needs. Parents may also become too involved, requesting too many changes, so that important improvements get lost with other changes that are being suggested. Instructors will also find that their work is increased; they must work to first understand the outcome, then build a curriculum around each outcome they are required to meet. Instructors have found that implementing multiple outcomes is difficult to do equally, especially in primary school. Instructors will also find their work load increased if they chose to use an assessment method that evaluates students holistically.

UNIT 2

NUMBER AND NUMERATION SYSTEMS

 

The idea of counting seems like such a simple concept, but when broken down, there are actually several distinct counting principles that progressively build toward a child being able to effectively count a group of objects.   There are five long-established counting principles that children must know in order to be able to count well. These five counting principles are:

1. Stable Order

Understanding the verbal sequence of counting; being able to say the number names in sequential order.

The first principle of counting involves the student using a list of words to count in a repeatable order. This ordered or “stable” list of counting words must be at least as long as the number of items to be counted.

For example, if a student wants to count 20 items, their stable list of numbers must be to at least 20.

Thinking deeper about stable order, we might consider rote counting from 0, counting on from a number (i.e.: “start at 6 and count to 18”) and counting backwards (i.e.: “count backwards from 15”) skills that are related to stable order.

Assisting students in acquiring these skills and flexibility to count on and count backwards will take time, but is helpful to building a deep understanding of counting and quantity.

Strategies that Support Student Learning

· Putting pictures of items in order from smallest to largest, based on quantity, and counting them forwards and backwards.

· Organizing objects in order, without numbers at first, then adding the symbols later.

If you observe:

· A child miscounting orally by rote or with objects…

Consider:

· Intentionally miscounting and ask the child to tell you what number you missed.

2.  ONE-TO-ONE CORRESPONDENCE

Understanding that when saying the names of the numbers in sequence, each object receives one count and one only one count.

Understanding that each object in a group can be counted once and only once. It is useful in the early stages for children to actually tag or touch each item being counted and to move it out of the way as it is counted.

In order for students to understand and apply the one-to-one counting principle, they must be able to orally count by rote.

We can promote the development of this skill by regularly counting items during play and everyday life. Encouraging students to show quantities on their fingers is also helpful.

Strategies that Support Student Learning

· Encouraging students to “tag” or move items out of the way while counting.

· Matching items with pictures. For example, using search and find books.

· Encouraging students to create a tally chart to count and track the quantity of food, toys, sounds (i.e.: taps on a drum), letters in a word or words in a sentence.

If you observe:

· A child playing in the kitchenette area preparing food for stuffed animals…

Consider:

· Asking how many items of food they are preparing or how many people they are cooking for.

3. CARDINALITY

Understanding that the last number spoken in a counting sequence names the quantity for that set.

Understanding that the last number used to count a group of objects represents how many are in the group.

A child who recounts when asked how many candies are in the set that they just counted, may not have an understanding of the cardinality principle.

If you are struggling to assess whether a student firmly grasps the cardinal principle, consider asking the student to count a group of items and then ask them to put the same quantity into a bag. If they must recount, they may not have a firm understanding of cardinality.

Strategies that Support Student Learning

· Encouraging students to show you a group of items to match a specific number.

· Ask students to count a group of items in a set. Then, explicitly ask them to show you how many objects in that group represent that amount.

If you observe:

· A child building a tower out of lego…

Consider:

· Asking if they can use the same amount of lego to create a path.

4. ABSTRACTION

Understanding that it doesn’t not matter what you count, how we count stays the same. For example, any set of objects can be counted as a set, regardless of whether they are the same color, shape, size, etc. This can also include non-physical things such as sounds, imaginary objects, etc.

Abstraction requires an understanding that we can count any collection of objects, whether tangible or not.

For example, the quantity of five large items is the same count as a quantity of five small items or a mixed group of five small and large things.

Another example may include a student being able to count linking cubes that represent some other set of objects like cars, dogs, or bikes.

Children often consider groups of larger items to have more value than groups of smaller items. For example, a child may believe that the quantity of the 3 cars in the parking lot is larger than the 3 toy cars placed on the play mat

Strategies that Support Student Learning

· Counting non-tangible quantities such as sounds, actions, words, questions or steps.

· Matching groups of different items with the same quantity.

If you observe:

· A child playing with toys of different sizes…

Consider:

· Taking a group of 2 larger items and a group of 3 smaller items and asking which has more.

 

5. ORDER IRRELEVANCE

Knowledge that the order that items are counted in is irrelevant—left-to-right, right-to-left, in a random fashion—as long as every object in the set is given one count and only one count Gelman and Galistel, 1978

The order in which items are counted is irrelevant.

Students have an understanding of order irrelevance when they are able to count a group of items starting from different places. For example, counting from the left-most item to the right-most and vice versa.

While the order irrelevance principle may seem obvious to adults, many students hold the misconception that the order you count objects does matter until as late as grade 4. Explicitly teaching this principle is important. It should also be noted that just because a child is strong in this principle, they may still be weak in other counting principles.

Strategies that Support Student Learning

· Counting sets of items from left-to-right, right-to-left, top-to-bottom and bottom-to-top.

· Counting sets of unique items (different colour, shape, etc.) in a variety of orders.

If you observe:

· A child counting a set of toy cars…

Consider:

· Asking if they can predict how many cars there would be if they started counting from a different spot.

6. CONSERVATION

Understanding that the count for a set group of objects stays the same no matter whether they are spread out or close together.

If a student counts a group of items that are close together and then needs to recount after you spread them out, they may not have developed an understanding of the principle of conservation.

 

Counting and Quantity Tip: Avoid Rushing to Symbols

It is all too common that we rush towards symbols in mathematics and counting is no different. Help children develop a firm grasp of the quantity associated with each number concretely before we formally introduce the symbolic form of number.

7. SUBITIZING

In general, subitizing is the ability to “see” or visualize a small amount of objects and know how many there are without counting. While this idea may seem simple on the surface, subitizing is actually quite complex. If we dig deeper, we can see that there are two types of subitizing that could be going on in our mind when we are learning to count called perceptual subitizing and conceptual subitizing.

Perceptual Subitizing

Perceptual subitizing takes place when you are able to look at a group of objects and know how many objects there are without having to do any thinking. Often times, when we look at groups of 5 objects or less, we are subitizing perceptually.

Examples of perceptual subitizing could include:

· Knowing there are 3 candies on a table without counting the candies.

· Knowing you rolled 5 with a single die without counting the dots.

· Knowing there are 2 cars in your driveway without counting the cars.

Conceptual Subitizing

Conceptual subitizing takes place when you are still able to “see” how many objects are in group, but the number of objects is too large to subitize without decomposing into two or more smaller groups.

We often shift from perceptual subitizing to conceptual subitizing when the number of objects in a group is larger than 5.

You may find that you are able to perceptually subitize groups of more than 5 items when the items are organized in familiar ways. For example, most “know” they have rolled 6 on a single die because of the familiar arrangement of the dots. However, you may struggle to perceptually subitize those 6 dots if they were arranged in an unfamiliar way and resort to conceptually subitizing by breaking up the 6 dots into two groups of 3 in your mind without even realizing it!

While many may believe that using 5- and 10-frames in early mathematics is simply because 5’s and 10’s are very friendly numbers in our base 10 system, we can see that the organization of items in a 5-frame can be an important tool to help students shift from perceptual subitizing to conceptual subitizing.

 

Strategies that Support Student Learning

· Ask children to count how many objects are in a set that is out of reach or difficult to physically tag using one-to-one correspondence (i.e.: cars in the driveway, chairs at the table, etc.).

· Create dot cards using pieces of paper with small quantities of dots on each, arranged in different configurations and play matching games, war and other fun card games with them.

If you observe:

· A child playing with a small quantity of items…

Consider:

· Asking them how many items they are playing with.

8. HIERARCHICAL INCLUSION

Understanding that all numbers preceding a number can be or are systematically included in the value of another selected number.

For example, knowing that within a group of 5 items, there is also a group of 4 items within that group; 3 items within that group; 2 items… and so on.

Hierarchical inclusion is an important landmark that students must reach in order to fully understand cardinality and to begin composing numbers (i.e.: composing a set of 5 items by combining a set of 2 items with a set of 3 items) and decomposing numbers (i.e.: decomposing a set of 6 items by separating into a set of 4 items and a set of 2 items).

Strategies that Support Student Learning

· Ask children to name the number that is “one larger” or “one smaller” than a number.

· Ask for a number that is “inside” the number 7.

9. MOVEMENT IS MAGNITUDE

Understanding that as you move up the counting sequence (or forwards), the quantity increases by one and as you move down (or backwards), the quantity decreases by one or whatever quantity you are going up/down by.

 

Despite many believing that learning the alphabet is the same as learning how to count, our number system is much more complex than the alphabet. The letters in the alphabet are arranged in an order for no particular reason, whereas the stable list of numbers we use for measuring quantity through counting is ordered by magnitude.

 

10. Unitizing

Unitizing refers to the understanding that you can count a large group of items by decomposing the group into smaller, equal groups of items and then count those.

For example, if there is a large group of candies on a table, one might choose to create groups (or “units”) of 2 (often doing this by perceptually subitizing these groups) and skip counting up by 2’s. Some may choose to create “units” of 3 and skip count up by 3’s.

We might consider connecting unitizing back to 1-to-1 correspondence by thinking of unitizing groups of 2 as “2-to-1 correspondence” in the situation where we are counting 2 candies for every 1 group or unitizing groups of 3 as “3-to-1 correspondence” in the situation where we are counting 3 candies for every 1 group. I find by thinking of unitizing this way, it can begin revealing connections to multiplication and the underlying ratios that exist whenever we count any quantity.

Understanding Place Value Through Unitizing

Unitizing is also important for students to understand that objects are grouped into tens in our base-ten number system. For example, once a count exceeds 9, this is indicated by a 1 in the tens place of a number.

UNIT 3

PLACE VALUE 10 TO 10,000 BASIC OPERATIONS ON WHOLE NUMBERS AND FACTS

 

DEVELOPING PLACE VALUE CONCEPTS: 10 TO 10,000

Number and place value are foundational concepts for all mathematics learning. This means we need to address how to teach place value as early as possible so that pupils can secure their knowledge of the concept.

How do you develop an early understanding of place value in the primary school classroom? Let’s start by defining place value. It is a system for writing numerals where the position of each digit determines its value. Each value is a multiple of a common base of 10 in our decimal system.

Here are some teaching strategies I’ve found useful when helping learners develop an early understanding of place value.

Progress through concepts systematically

Developing an understanding of place value requires systematic progression. Each new concept should build on previous learning experiences so that pupils can gain deeper, relational understanding as they go.

This approach ensures knowledge is developed, refined and applied correctly as numbers become meaningful tools for solving problems rather than just a series of symbols on a page. Most importantly, this starts our learners on the path to becoming confident problem solvers and pattern spotters.

Use the CPA approach to establish meaning

The CPA (Concrete, Pictorial, Abstract) approach helps pupils connect a physical representation of a number (concrete manipulatives) to that same quantity as shown in drawings or graphics (pictorial), and finally to the actual written name and symbol for that number (abstract).

I view concrete resources as meaning makers. They add meaning to abstract representations of numbers so that when learners progress to the abstract phase, they know what those numbers stand for, what they mean, and how they relate to each other.

If a pupil can identify the meaning of each component in a problem, they are far more confident in how they work to solve it.

Teach the ‘ten-ness of ten’

‘Ten’ is the foundational building block of our Base 10 numeration system. At an early level, spend as much time as possible studying the numbers from 0 to 10, as understanding the ‘ten-ness of ten’ is crucial for maths attainment, and it cannot be rushed.

Once this understanding is locked-in, follow this with an introduction to number bonds. Start with the additive relationships between numbers less than 10, then progress to adding and subtracting up to 10. This ensures that learners see 10 as an important ‘base’ number in all of their future maths applications.

Progress to 20, then to 40

I make sure to take my time teaching ten and teen numbers so that a solid understanding of place value with numbers up to 20 is properly established.

I then extend the place value concept by working with numbers up to 40 — followed by addition and subtraction to 40.

Because pupils have learned to make 10 and use number bonds, they are ready to begin working with multi-digit numbers and regrouping. Focusing on numbers to 40 while developing the concept of place value also allows learners to associate numbers with easily-managed, physical quantities (meaning makers).

Use base 10 blocks for 100 and 1000

The work we’ve done building a gradual understanding of place value will have prepared pupils to progress to three-digit numbers. So we can now move on to studying up to 100.

We start here by developing an understanding of numbers in multiple place value representations. For example, one thousand five hundred is 15 hundreds or 150 tens.

Once they get the hang of that, learners then sharpen their counting, reading, and writing skills for numbers up to 1,000. Moving into addition and subtraction with numbers up to 1,000 — with and without regrouping — is the next step.

Here is where our work establishing an early understanding of place value is key, because pupils will intrinsically know why these algorithms work for three and four-digit numbers. Base 10 blocks are a great tool to help solidify those earlier place value ideas when working with numbers up to the thousands.

Approach larger numbers the same way

The CPA approach is once again our answer to learning place value in larger numbers. Apply those skills and always be on the lookout for chances to extend number and place value concepts.

For example, you can identify and complete number patterns or find missing digits on a number line.

From there you can explore strategies for mental mathematics as well as addition and subtraction for numbers up to 10,000. Take learners even deeper by having them explore place value with an emphasis on multiplication, division, and decimals.

UNIT 4

CLASSROOM ASSESSMENT IN MATHEMATICS IN THE JHS1-3

 

GUIDELINES FOR ASSESSING STUDENT LEARNING

Effective assessment design enhances student learning and engages students with different learning styles. Assessment influences what a student interprets to be the important learning goals for a course. The assignments we design constitute the means by which we assess student learning. In the best-case scenario, assessments should be aligned with course goals and objectives. The following guidelines will help you to design assessments that promote your students’ learning.

Assessment Influences the Process of Learning

1. Provide spaced assessed tasks to enable students to allocate sufficient time to study over a suitable time period and avoid "cramming."

· Students submit drafts/works in progress or display work publicly at specific points during the course.

· For group work, provide mechanisms for formalized peer feedback throughout the project.

Design frequent tasks rather than one end of course assessment (or build in steps).

2. Design the assessment so that students tackle the task appropriately, i.e., they engage in the process of learning rather than simply producing a final product.

· Allocate some percentage of the overall mark to drafts/works in progress or justifications of decisions made while completing the assignment.

· Allow students to reveal their errors and explain their corrections to validate the process of learning.

Value the learning process rather than only the final assignment by allocating marks to students’ self-analysis of their learning.

3. Give students the opportunity to practice the skills they need for each assessment.

· Clearly explain the assessment criteria.

· Give feedback on formative work.

· Discuss the assessment task with students.

· Continue to improve your assessment tasks for future iterations of your course based on feedback and based on how effective they were (or not) in enhancing students’ learning.

Use Feedback to Enhance Learning

4. Provide sufficient and detailed feedback.

· Use feedback and self-assessment sheets.

· Consider using audio or video recordings (e.g., for student presentations) and provide opportunities for students to discuss their reflections on their presentations and how they would like to improve.

· Avoid checks and crosses, or less meaningful terms like “Great work!” or “Poor.”

5. Focus your feedback on student performance, learning, or actions the student can control.

· Identify errors clearly.

· Outline options for action that are reasonable.

· Avoid personal comments which can reduce a student’s sense of competence (linked to motivation).

6. Provide timely feedback, or feedback that is given while it matters to the student and can be used to improve future performance.

· Discuss model answers or exemplars immediately after students submit their work, while the ideas are fresh in their minds.

· Use peer feedback: immediate peer feedback is preferable to late professor feedback.

· Computer-based practice tests (e.g., multiple choice tests on Canvas) can provide immediate feedback for student self-paced study; however, ensure that explanations for incorrect and correct answers are provided. Also, provide “thought questions” to encourage students to consider the critical aspects of the questions (and discourage them from randomly guessing).

7. Align feedback with the learning goals of the assignment and the assessment criteria.

· Align your feedback with the aim of the assignment: are you trying to increase interest and motivation with new students, or promote reflective learning, or identify and correct misconceptions?

· Use self and peer assessment to encourage critical thinking and internalization of assessment criteria and standards.

8. Provide feedback that is appropriate to the student’s breadth and depth of background, experience, and level of independence.

· Feedback needs to be understandable to the student and provide ways for the student to progress to the next stage or level of understanding.

· Provide feedback sensitive to the student’s understanding of the discipline involved (i.e., separate generic study skills feedback from discipline-specific comments).

9. Feedback needs to be read and noticed.

· Provide ways for students to have an active role in eliciting useful feedback. For example, you can have them identify and list points they need feedback on.

· Where appropriate, it helps to give feedback only (no grade).

· Use self-assessment prior to any official grading.

· Use two-part assignments: formative feedback at part 1; grade only at part 2.

· Use self-assessment, professor feedback, and then supply a grade.

10. Feedback is acted on by the student.

· Follow up the feedback and be encouraging.

· Provide “feed-forward” (applies to future work).

· Use feedback to promote self-directed learning.

 

EFFECTIVE ASSESSMENT

Assessments of all types provide evidence for the practitioner to make decisions, often in collaboration with the learner, about the next steps forward in the learning program.

ASSESSMENT TASKS

Assessments may be formal or informal and they may be formative or summative. Assessment tasks vary from informal questions during a learning activity to a formal written tests at the end of a learning program. Assessments of all types provide evidence for the practitioner to make decisions, often in collaboration with the learner, about the next steps forward in the learning program.

​​​​​FORMATIVE ASSESSMENT

Practitioners engage in both formal and informal assessment as learners progress along the learning continuum. Much informal assessment occurs during a class or group session when practitioners ask questions of individual learners attempting a learning activity and when they engage the group in discussion or ask them to perform an action, for example retrieve a file or throw a ball.

Practitioners undertake informal assessments to understand how well the learner is progressing towards achieving the learning intentions and success criteria, and the assessment is often tailored to the individual learner. These formative assessments provide the practitioner with evidence of the learner’s progress and concepts, knowledge and skills not yet understood. The practitioner uses this evidence to adjust the learning program to meet the learner’s needs.

Formative assessments may be conducted in a more formal manner. Formal assessments are often written tasks that require the learner to respond in a particular way, for example to write an essay, perform a dance, or create a movie. The response will be assessed according to a rubric or marking scheme developed against the success criteria.

A common type of formal assessment is the written test. Writing effective written tests is a whole topic in itself and advice about these will be provided in the coming months. Tests are usually timed assessments and may comprise multiple choice, short answer, and extended answer questions sometimes in response to case studies or scenarios. The practitioner selects particular types of tests and questions depending on the purpose of the assessment, the depth of response required and how quickly they wish to give feedback. Multiple choice tests can be marked quickly and feedback given almost immediately but tests requiring extended responses take longer to mark and the feedback will be slower in reaching the learners.

SUMMATIVE ASSESSMENT

Summative assessments are often developed as formal assessment tasks that provide evidence of the learner’s mastery of knowledge, skills and understandings at a point in time. They measure what the learner has achieved against the achievement standards. The practitioner may use summative assessments for reporting to the learner and their parents about the learner’s achievement.

Whilst a summative assessment provides evidence of a learner’s achievement at a point in time, it can also be regarded as formative assessment since the evidence indicates what a learner has mastered and what knowledge, skills and understandings they still need to learn. As summative assessment usually occurs at the end of a learning program, unit or semester, the evidence can be provided to the next practitioner to work with the learner so that they will understand where the learner is on the learning continuum. They can then plan a more appropriate learning program.

QUALITIES OF EFFECTIVE FORMAL ASSESSMENT TASKS

Practitioners may develop their own formal assessment tasks that are specific to their learning domain and the context in which they are teaching, for example assignments, role plays, and simulations. It should be noted that when practitioners engage learners in co-construction of an assessment task learners are more likely to take ownership of their learning.

Effective assessment tasks are transparent and co-constructed so the learner knows the purpose of the task, what is expected and how the task will be assessed.

The type of assessment task set depends on the purpose of the task. Sometimes there is an emphasis on tasks that are authentic, open-ended and require deep understanding of an area of content. In other circumstances administering a simple multiple choice assessment will provide the practitioner with useful information. An effective assessment is always appropriate to its purpose and able to be readily administered by the practitioner.  In selecting an appropriate assessment, consideration is given to these characteristics: reliability, validity, inclusivity, objectivity and practicality.

EFFECTIVE FORMAL ASSESSMENT TASKS

Practitioners need access to a wide repertoire of assessment tasks to gather evidence of the different forms of learning across the curriculum. Increasingly as learning encourages more open-ended aspirations, tasks need to be developed that are fit for the purpose of gathering information about a wider variety of skills and understandings, for example critical and creative thinking and collaboration.

Practitioners provide learners with the opportunity to demonstrate their knowledge, skills and understanding if the assessment tasks:

· Directly relate to the learning intentions or particular learning outcome

· Are explicit about what learners are required to do

· Are time efficient and manageable

· Include clear and explicit assessment criteria

· Provide challenge for the full range of learners being assessed

· Are fair to all students including those with additional needs

· Are scored or marked based on transparent rubrics

· Are appropriate to where learners are in their learning

ASSESSMENT CRITERIA

Learners can effectively demonstrate what they know, understand and can do if they are provided with, or collaboratively develop with the practitioner, the assessment criteria for an assessment task. Effective assessment criteria:

· Are known to the learners​​

· Are clear and explicit

· Focus on the important criteria and substance of the task (not every tiny detail)

· Allow learners to achieve at a high level

· Provide for a range of quality in the work

ASSESSMENT MATERIALS

Informing learners about the materials or activities they are expected to submit for an assessment task ensures they have the opportunity to demonstrate their knowledge, skills and understanding in the form expected by the practitioner and that all elements of a task are completed. Learners should be provided with:

· Stimulus material, case study, problem

· Questions/activities to be completed

· Assessment criteria or rubric​

· List of what must be submitted​​

DESIGNING EFFECTIVE ASSESSMENT TASKS

An assessment task is a tool, device or constructed situation that creates the opportunity for learners to demonstrate or display the nature and depth of their learning.

Effective teachers design assessment tasks that require students to demonstrate knowledge and skills at man levels.  Tasks will include lower order processes like comprehension, and higher order processes like synthesis and evaluation.

· When teachers explain the connections between learning goals, learning activities and assessment tasks, then the students can use learning goals to monitor and progress their learning. 

· Assessments should be:

· Authentic, fit for purpose and reflect the learning program and objectives.

· Aligned to curriculum achievement standards.

· Integrated into a learning sequence.

Assessment tasks should include a  range of formative and summative assessment strategies, and teachers will be able to clearly explain the connections between learning goals, learning activities and assessment tasks so that students can use learning goals to monitor and progress their learning.

ASSESSMENT TECHNIQUES

Effective questioning

Much assessment occurs during classroom interactions between practitioner and learners. The quality of questions asked by the teacher and learners, the depth of answers supplied by learners, the quality of class discussions and the detailed observations practitioners make of learners at work all provide evidence of learning including shallow or deep understanding and misconceptions.

Questioning is quick, effective method for gathering evidence of learners' understanding of ideas, knowledge and concepts and skills to be applied. Effective questions encourage learners to think more deeply and provide the practitioner with greater insight into the level of understanding of whole groups and individuals.  The practitioner can quickly adjust their practice to meet the learner's needs as identified through using effective questioning techn​iques.​

Effective questioning techniques

​Learners' responses to questions give the practitioner feedback about their level of understanding if the questions are open-ended and formed to elicit informative responses.​

A more comprehensive discussion about the types of questions that encourage learners to think and reveal their level of understanding is found in the McComas and Abraham paper, Asking more effective questions.

Digital portfolios, or learning journals in whi​ch learners record their learning goals and learning experiences provide the practitioner with useful evidence about what learners understand and what skills and knowledge they believe they need to develop.

RUBRICS FOR ASSESSMENT

A rubric is an explicit set of criteria used for assessing a particular type of work or performance (TLT Group, n.d.) and provides more details than a single grade or mark. Rubrics, therefore, will help you grade more objectively.

Have your students ever asked, “Why did you grade me that way?” or stated, “You never told us that we would be graded on grammar!” As a grading tool, rubrics can address these and other issues related to assessment: they reduce grading time; they increase objectivity and reduce subjectivity; they convey timely feedback to students and they improve students’ ability to include required elements of an assignment (Stevens & Levi, 2005). Grading rubrics can be used to assess a range of activities in any subject area

ELEMENTS OF A RUBRIC

Typically designed as a grid-type structure, a grading rubric includes criteria, levels of performance, scores, and descriptors which become unique assessment tools for any given assignment. The table below illustrates a simple grading rubric with each of the four elements for a history research paper. 

CRITERIA

Criteria identify the trait, feature or dimension which is to be measured and include a definition and example to clarify the meaning of each trait being assessed. Each assignment or performance will determine the number of criteria to be scored. Criteria are derived from assignments, checklists, grading sheets or colleagues.

EXAMPLES OF CRITERIA FOR A TERM PAPER RUBRIC

· Introduction

· Thesis

· Arguments/analysis

· Grammar and punctuation

· Spelling

· Internal citations

· Conclusion

· References

LEVELS OF PERFORMANCE

Levels of performance are often labeled as adjectives which describe the performance levels. Levels of performance determine the degree of performance which has been met and will provide for consistent and objective assessment and better feedback to students. These levels tell students what they are expected to do. Levels of performance can be used without descriptors but descriptors help in achieving objectivity. Words used for levels of performance could influence a student’s interpretation of performance level (such as superior, moderate, poor or above or below average).

EXAMPLES TO DESCRIBE LEVELS OF PERFORMANCE

· Excellent, Good, Fair, Poor

· Master, Apprentice, Beginner

· Exemplary, Accomplished, Developing, Beginning, Undeveloped

· Complete, Incomplete

· Yes, No

Levels of performance determine the degree of performance which has been met and will provide for consistent and objective assessment and better feedback to students.

SCORES

Scores make up the system of numbers or values used to rate each criterion and often are combined with levels of performance. Begin by asking how many points are needed to adequately describe the range of performance you expect to see in students’ work. Consider the range of possible performance level.

EXAMPLE OF SCORES FOR A RUBRIC

1, 2, 3, 4, 5 or 2, 4, 6, 8

DESCRIPTORS

Descriptors are explicit descriptions of the performance and show how the score is derived and what is expected of the students. Descriptors spell out each level (gradation) of performance for each criterion and describe what performance at a particular level looks like. Descriptors describe how well students’ work is distinguished from the work of their peers and will help you to distinguish between each student’s work. Descriptors should be detailed enough to differentiate between the different level and increase the objectivity of the rater.

Descriptors...describe what performance at a particular level looks like.

DEVELOPING A GRADING RUBRIC

First, consider using any of a number of existing rubrics available online. Many rubrics can be used “as is.” Or, you could modify a rubric by adding or deleting elements or combining others for one that will suit your needs. Finally, you could create a completely customized rubric using specifically designed rubric software or just by creating a table with the rubric elements. The following steps will help you develop a rubric no matter which option you choose.

1. Select a performance/assignment to be assessed. Begin with a performance or assignment which may be difficult to grade and where you want to reduce subjectivity. Is the performance/assignment an authentic task related to learning goals and/or objectives? Are students replicating meaningful tasks found in the real world? Are you encouraging students to problem solve and apply knowledge? Answer these questions as you begin to develop the criteria for your rubric.

Begin with a performance or assignment which may be difficult to grade and where you want to reduce subjectivity.

2. List criteria. Begin by brainstorming a list of all criteria, traits or dimensions associated task. Reduce the list by chunking similar criteria and eliminating others until you produce a range of appropriate criteria. A rubric designed for formative and diagnostic assessments might have more criteria than those rubrics rating summative performances (Dodge, 2001). Keep the list of criteria manageable and reasonable.

3. Write criteria descriptions. Keep criteria descriptions brief, understandable, and in a logical order for students to follow as they work on the task.

4. Determine level of performance adjectives. Select words or phrases that will explain what performance looks like at each level, making sure they are discrete enough to show real differences. Levels of performance should match the related criterion.

5. Develop scores. The scores will determine the ranges of performance in numerical value. Make sure the values make sense in terms of the total points possible: What is the difference between getting 10 points versus 100 points versus 1,000 points? The best and worst performance scores are placed at the ends of the continuum and the other scores are placed appropriately in between. It is suggested to start with fewer levels and to distinguish between work that does not meet the criteria. Also, it is difficult to make fine distinctions using qualitative levels such as never, sometimes, usually or limited acceptance, proficient or NA, poor, fair, good, very good, excellent. How will you make the distinctions?

It is suggested to start with fewer [score] levels and to distinguish between work that does not meet the criteria.

6. Write the descriptors. As a student is judged to move up the performance continuum, previous level descriptions are considered achieved in subsequent description levels. Therefore, it is not necessary to include “beginning level” descriptors in the same box where new skills are introduced.

7. Evaluate the rubric. As with any instructional tool, evaluate the rubric each time it is used to ensure it matches instructional goals and objectives. Be sure students understand each criterion and how they can use the rubric to their advantage. Consider providing more details about each of the rubric’s areas to further clarify these sections to students. Pilot test new rubrics if possible, review the rubric with a colleague, and solicit students’ feedback for further refinements.

TYPES OF RUBRICS

Determining which type of rubric to use depends on what and how you plan to evaluate. There are several types of rubrics including holistic, analytical, general, and task-specific. Each of these will be described below.

Holistic

All criteria are assessed as a single score. Holistic rubrics are good for evaluating overall performance on a task. Because only one score is given, holistic rubrics tend to be easier to score. However, holistic rubrics do not provide detailed information on student performance for each criterion; the levels of performance are treated as a whole.

· “Use for simple tasks and performances such as reading fluency or response to an essay question . . .

· Getting a quick snapshot of overall quality or achievement

· Judging the impact of a product or performance” (Arter & McTighe, 2001, p 21)

Analytical

Each criterion is assessed separately, using different descriptive ratings. Each criterion receives a separate score. Analytical rubrics take more time to score but provide more detailed feedback.

· “Judging complex performances . . . involving several significant [criteria] . . .

· Providing more specific information or feedback to students . . .” (Arter & McTighe, 2001, p 22)

Generic

A generic rubric contains criteria that are general across tasks and can be used for similar tasks or performances. Criteria are assessed separately, as in an analytical rubric.

· “[Use] when students will not all be doing exactly the same task; when students have a choice as to what evidence will be chosen to show competence on a particular skill or product.

· [Use] when instructors are trying to judge consistently in different course sections” (Arter & McTighe, 2001, p 30)

Task-specific

Assesses a specific task. Unique criteria are assessed separately. However, it may not be possible to account for each and every criterion involved in a particular task which could overlook a student’s unique solution (Arter & McTighe, 2001).

· “It’s easier and faster to get consistent scoring

· [Use] in large-scale and “high-stakes” contexts, such as state-level accountability assessments

· [Use when] you want to know whether students know particular facts, equations, methods, or procedures” (Arter & McTighe, 2001, p 28) 

Summary

Grading rubrics are effective and efficient tools which allow for objective and consistent assessment of a range of performances, assignments, and activities. Rubrics can help clarify your expectations and will show students how to meet them, making students accountable for their performance in an easy-to-follow format. The feedback that students receive through a grading rubric can help them improve their performance on revised or subsequent work. Rubrics can help to rationalize grades when students ask about your method of assessment. Rubrics also allow for consistency in grading for those who team teach the same course, for TAs assigned to the task of grading, and serve as good documentation for accreditation purposes. Several online sources exist which can be used in the creation of customized grading rubrics; a few of these are listed below.

UNIT 5

MICRO LESSONS AND USE OF TECHNOLOGY ACROSS JUNIOR HIGH SCHOOL NUMERACY

 

IMPORTANCE OF LESSON PLANNING

1. Inspiration. A thorough lesson plan inspired the teacher to improve the lesson plan further. You can make it better for the purpose of achieving the lesson plan in a better way.

2. Evaluation. A lesson plan helps the teacher to evaluate his teaching and to compare it with set objectives. This evaluation will help you in achieving the set targets in a better way .

3. Self-confidence. These lesson plans develops self-confidence in the teacher and make them to work towards definite goal.

4. Previous Knowledge of the Students. A teacher can take a proper care by considering the level and previous knowledge of the students in your class.

5. Organized Matter. A teacher will be able to finish a particular lesson in a limited time frame. This will help him or her to make the students learn a better and precise manner.

6. Ask Questions. A teacher will be able to ask proper and important questions to the students in the classroom. This will engage the students in communication and help them in retaining the lesson.

7. Guidance.  A lesson plan works as a guide for the teacher in the classroom. It tells you what to teach so that they can cover the entire lesson within a limited time frame.

8. Interest. A lesson plan creates the interest of the students in the lesson and makes them learn with curiosity in subject matter.

9. Stimulation. A lesson plan stimulates the teacher to think in an organized way. This helps you to match the ideal standard of teaching more quickly than ever.

10. Understand the Objectives. Through a lesson plan, a teacher is able to understand the objectives of the lesson properly and make his students to understand them too, with ease.

MICRO LESSON PLANNING

Simply put, a micro lesson plan is focused on one specific subject to be explored within a learning platform. Your micro lesson plan should be designed with the intention of carrying out short, succinct lessons for your learners to master. 

Within each lesson, there’s an opportunity to introduce a wide range of built-in features that result in higher engagement rates, retention rates, and most importantly, better learning experiences. One of these elements includes spaced repetition or Brain Boost spaced repetition app, where interactive lessons are automatically created based upon core, previously-learned content from the EdApp platform. This feature is backed by a highly-regarded Supermemo SM-2 interval algorithm and can be implemented for any EdApp user, aiming to instil key concepts and boost learners’ retention on any given topic. Explore the diagram below for a deeper look into EdApp’s many built-in features and elements found within a powerful Authoring Tool, all of which are designed to help you create the best micro lesson plan for your learners.

EdApp’s data-driven Authoring Tool gives you the opportunity to design beautiful, engaging, and effective micro lessons with a built-in tool – no coding required. The user-friendly platform hosts unlimited possibilities of multimedia, meaning you can easily embed videos, images, audio files, and external URLs into EdApp’s micro lessons easily and effectively.

EdApp’s extensive Template Library also includes over 50 intuitively designed templates that suit every type of learner. Choose from interactive templates that include multiple-choice, elements of gamification, conceptual formats, surveys, and more. EdApp’s cloud-based Translation Tool enables you to translate your courseware into over 100 languages with a few clicks and in record time. 

And this is just scratching the surface of EdApp’s capabilities, all of which are completely free and available for you to create the ultimately micro lesson plan to drive better learning results for you and your teams. 

HOW TO MAKE A LESSON PLAN FOR MICRO TEACHING IN 4 STEPS

1. Introduce learners to the topic with a title slide

It’s important to start by telling your learners what the lesson is about. This puts them in the right frame of mind by getting them thinking about learning and making them more receptive to new information. By giving them an overview of the topic, you’re also making them think about context (for more on why this is important, see this article on chunking strategy) and what they know about the subject already. The more they can relate to the subject matter, the more effective the lesson will be, all while providing a greater opportunity for the new information to embed into your learners’ long-term memory with ease. Pro tip: A simple introductory slide will do the job, but avoid presenting a wall of text as it will turn people off.

2. Begin knowledge transfer with video, text or both

Video is proven to be one of the best forms of knowledge transfer. Text is fine but, again, we recommend keeping it minimal to avoid turning off your learner. Using five-or-six content slides in a row, we find, is too many for a micro lesson plan. If all of the information is important, consider splitting the information into multiple lessons. We find using four slides is an optimal number for directed-focus lessons.

You can easily find perfect examples of a balance of video, text, and both in our Editable Content Library, where leading companies around the world have contributed high-quality, practical courseware – all for free.

For example, the United Nations Institute for Training and Research (UNITAR) has created a collection of brilliant lessons, including Meet the Sustainable Development Goals. Accessible now for all to take, this course introduces the 2030 Agenda for Sustainable Development and optimal understanding of the Sustainable Development Goals (SDGs). This learning content guides learners through a multi-part course to share amongst colleagues, friends, and family, for effective mainstreaming of the SDGs. Curious to discover the content for yourself?

3. Micro teach to reinforce content using interactive questions and games

Next, our learning experts recommend introducing interactive questions to help reinforce your content. If your learners get the answer right, you can reinforce why the right answer was important. However, if they get it wrong, it’s very important to quickly correct any misconception and explain what the right answer is: any delay will increase the likelihood of retaining the wrong information. Just as you would for answering the question correctly, it’s important to tell your learners what the correct answer is and why the information is important – this will leave them with a lingering takeaway message. EdApp boasts an expansive library of over 50 templates to help you create interactive questions or games to help engage your learners and reinforce your concepts.

4. Applying gamification to a micro lesson plan

Playing games makes for effective learning, but making your lessons competitive (and even rewarding) will drive effectiveness even further. There are various microteaching methods for doing this (which ultimately depends on which Learning Management System you use) but scoring answers, setting time limits and awarding points for completing tasks within the lesson all increase learners’ engagement. 

EdApp’s built-in rewards program is built around earning ‘stars’. Based on your preferences, you can easily reward your learners through stars, which can then be turned into real rewards like a Starbucks or Amazon gift card, for example. 

Gamification and real prizing entice learners to collect more stars, as they complete your microlessons. Be generous though – don’t give out one star after they’ve sat through 20 slides! Offering real prizes for best performance or simply completing a course on time naturally acts as a learning incentive.

A real-world example can be found in the realm of retail trainees: asking which statements about a product represent correct or incorrect things to say to customers – by swiping left for incorrect answers and swiping right for correct answers – gamifies the interactive learning, thereby improving retention. 

Micro learning with built-in gamification is an amazing tool to adopt for the success of your organization’s training strategy. We know that micro learning has proven successes and one of the best things about it, is that you have the freedom to implement the practice in many different ways, giving you the space to deploy custom, bespoke learning experiences to your audiences.

UNIT 6

INTEGERS

 

NUMBER SENSE

In mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations". Other definitions of number sense emphasize an ability to work outside of the traditionally taught algorithms, e.g., "a well-organized conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms".

Psychologists believe that the number sense in humans can be differentiated into the approximate number system, a system that supports the estimation of the magnitude, and the parallel individuation system, which allows the tracking of individual objects, typically for quantities below 4.

There are also some differences in how number sense is defined in math cognition. For example, Gersten and Chard say number sense "refers to a child's fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparisons."

In non-human animals, number sense is not the ability to count, but the ability to perceive changes in the number of things in a collection. All mammals, and most birds, will notice if there is a change in the number of their young nearby. Many birds can distinguish two from three.

Researchers consider number sense to be of prime importance for children in early elementary education, and the National Council of Teachers of Mathematics has made number sense a focus area of pre-K through 2nd grade mathematics education.[9] An active area of research is to create and test teaching strategies to develop children's number sense. Number sense also refers to the contest hosted by the University Interscholastic League. This contest is a ten-minute test where contestants solve math problems mentally—no calculators, scratch-work, or mark-outs are allowed.[10]

CONCEPTS INVOLVED IN NUMBER SENSE

The term "number sense" involves several concepts of magnitude, ranking, comparison, measurement, rounding, percents, and estimation, including:[11]

· Estimating with large numbers to provide reasonable approximations;

· Judging the degree of precision appropriate to a situation;

· Understanding the hidden meaning of numbers through analytical and critical thinking (e.g., freakonomics[12]);

· Rounding (understanding reasons for rounding large numbers and limitations in comparisons);

· Choosing measurement units to make sense for a given situation;

· Solving real-life problems involving percentages and decimal portions;

· Comparing physical measurements within and between the u.s. and metric systems; and

· Comparing degrees fahrenheit and celsius in real-life situations.[11]

· Those concepts are taught in elementary-level education.

NUMBER SENSE IS A GROUP OF SKILLS THAT ALLOW KIDS TO WORK WITH NUMBERS. IT INCLUDES SKILLS LIKE:

· Understanding quantities.

· Grasping concepts like more and less, and larger and smaller.

· Recognizing relationships between single items and groups of items (seven means one group of seven items).

· Understanding symbols that represent quantities (7 means the same thing as seven).

· Making number comparisons (12 is greater than 10, and four is half of eight).

· Understanding the order of numbers in a list: 1st, 2nd, 3rd, etc.

Some people have stronger number sense than others. But struggling with number sense can lead to challenges in school and in everyday life.

TROUBLE WITH NUMBER SENSE

People who struggle with number sense can have trouble with basic math operations. They may not understand what it means to add to or subtract from a group of items, for instance.

Imagine a pile of seven beads. If you take away two of them, students with poor number sense might not realize that the number of beads has shrunk. They might not recognize that subtracting the beads means the group of seven is now a group of five.

Likewise, if you add three beads to the pile, they might not realize the group of beads has gotten bigger. And they might not know that adding three to the pile of seven makes it a pile of 10.

Trouble with number sense can also make it hard to do multiplication and division. Students may not grasp that it’s simpler to combine items from several groups by multiplying them rather than by adding them. Or that division is the simplest way to break up groups into their component parts.

It can also make it hard to understand concepts like distance and time. That’s because these concepts rely on numerals to symbolize amounts. The same goes for measurement. The task of measuring requires a good understanding of the relationships between parts and wholes.

HOW THE SCHOOL CAN HELP

Kids can develop these key skills, but it takes time and practice. This makes it challenging for schools to work on number-sense skills the same way they work on specific reading, writing, and math skills.

When a child struggles with math, schools often focus first on reteaching the specific math skills being taught in class. The teacher might then ask the child to do extra worksheets. Or kids may use computer-based activities for extra practice.

This approach often doesn’t work for kids with weak number sense, though. In that case, schools usually turn to intervention through RTI or MTSS processes. With intervention systems, kids typically:

· Work with “manipulatives” like blocks and rods to understand the relationships between amounts.

· Do exercises that involve matching number symbols to quantities.

· Get a lot of practice estimating.

· Learn strategies for checking an answer to see whether it’s reasonable.

· Talk with their teacher about the strategies they use to solve problems.

· Get help correcting mistakes they make along the way.

For many kids with weak number sense, intervention is enough to catch up. But some kids need more support. They may need to be evaluated for special education to get the help they need.

HOW YOU CAN HELP YOUR CHILD AT HOME

If your child struggles with number sense, start with the basics.

Practice counting and grouping objects. Then add to, subtract from, or divide the groups into smaller groups to practice operations. You can also combine groups to show multiplication. Try matching numerals with quantities of objects, too.

Work on estimating. Build questions into everyday conversations, using phrases like “about how many” or “about how much.”

Talk about relationships between quantities. Ask your child to use words like more and less to compare things.

Build in opportunities to talk about things like time and money. For example, you could ask your child to keep track of how long it takes to drive or walk to the grocery store. Then compare it with how long it takes to get to school. Ask which takes longer.

Try not to jam all these activities into a short period of time. It takes time for kids to develop number sense, and you don’t want them to get frustrated. Work on the activities when it’s convenient, over a period of months. Repeat activities, but leave time in between.

FUNDAMENTAL OPERATIONS ON INTEGERS

We have four fundamental operations on integers. They are addition, subtraction, multiplication, and division.

1. Addition of integers

A monkey is sitting at the bottom in an empty water tank 8 ft high. The monkey wants to jump to the top of the water tank. He jumps 3 ft up and then slides 2 ft down. In how many jumps will the monkey reach the top of the empty water tank?

Monkey’s jump:

The monkey will reach the top of the empty water tank in the 6th jump.
Two integers can be added in the same way as two whole numbers are added but while adding negative integers, we have to move to the left on the number line.

Addition of integers having the same sign
1. The sum of two positive integers is the sum of their absolute values with a positive sign.

Example 1: Add (+ 6) + (+4).
Solution: On a number line, first draw an arrow from 0 to 6 and then go 4 steps ahead. The tip of the last arrow reaches +10. So, (+ 6) + (+ 4) = +10

2. The sum of two negative integers is the sum of their absolute values with negative sign(-).

Example 2: Add (-3) + (-4).
Solution: On a number line, first we draw an arrow on the left side of zero from 0 to -3 and then further move to the left 4 steps. The tip of the last arrow is at -7. So, (-3) + (-4) = (-7)

Addition of integers having opposite signs
The sum of two integers having opposite signs is the difference of their absolute values with the sign of integer of greater absolute value.

Example 3: Add(+6) + (-9).
Solution: On a number line, first we draw an arrow from 0 to 6 on the right and then go 9 steps to the left. The tip of the last arrow is at -3. So, (+6) + (-9) = (-3)

2. Subtraction of integers

In subtraction, we change the sign of the integer which is to be subtracted and then add to the first integer. In other words, if a and b are two integers, then a – b = a + (-b)

Example 4: Subtract 5 from 12.
Solution: (12) – (5) = (12) + (-5) = 7

Example 5: Subtract -7 from -15.
Solution: (-15) – (-7) = (-15) + (7)= -8

Example 6: Subtract 6 from -10.
Solution: (-10) -(6) = (-10) + (- 6)

Example 7: Subtract (-5) from 4.
Solution: 4 – (-5) = 4 + (5) = 9

To subtract (-5) from 4, we have to find a number which when added to (-5) gives us 4. So, on the number line we start from (-5) and move up to 4. Now find how many units we have moved. We have moved 9 units.
So, 4-(-5) =9

Note:

Addition of integers
(a) The sum of two integers with like signs is the sum of their absolute values with the same sign.
(b) The sum of two integers with unlike signs is the difference of their absolute values with _the sign of the greater absolute value.

Subtraction of integers
The sign of the integer is changed which is to be subtracted and then added to the first integer.

3. Multiplication of integers

Multiplication of integers having the same sign
When two integers have the same sign, their product is the product of their absolute values with positive sign.
Examples
(a) (+6) × (+7) = + 42 or 42
(b) (+5) × (+10) = + 50 or 50
(c) (-3) × (-5) = + 15 or 15
(d) (-20) × (-6) = 120
(e) (12) × (5) = 60

Multiplication of integers having opposite signs
The product of two integers having opposite signs is the product of their absolute values with negative sign.
Examples
(a) (-10) × (8) = (- 80)
(b) (- 5) × (7) = (-35)
(c) (12) × (-3) = (-36)
(d) (-6) × (3) = (-18)
(e) 5 × (-4) = (-20)

Note:
plus × minus = minus
minus × plus = minus
minus × minus = plus
plus × plus = plus

4. Division of integers

Division of integers having the same sign
Division of two integers having the same sign is the division of their absolute value with a positive sign. If both integers have the same sign, then the quotient will be positive.
Examples:
(a) (+9) ÷ (+3) = (3)
(b) (-9) ÷ (-3) = (3)
(c) (-24) ÷ (-12) = (2)

Division of integers having opposite signs
If both integers have different signs, the quotient will be negative.
Examples: (a) 12 ÷ (-3) = (-4)
(b) (-10) ÷ (5) = (-2)
(c) (-18) ÷ (3) = (-6)

Example 8: Evaluate (-13) – (-7 – 6).
Solution: (-13) – (-7 – 6)
= (-13) -(-13)
= (-13) + (13) (Opposite to each other) = 0

Example 9: Subtract (-5128) from 0.
Solution: 0 – (-5128) = 0 + 5128 = 5128

Example 10: Divide (4000) + (- 100).
Solution: 4000−100 = -40

Example 11: Multiply (-18) and (-8).
Solution: (-18) × (-8) = 18 × 8 = 144

Note:

Multiplication of integers
(a) When two integers have the same sign, their product is the product of their absolute values with a positive sign.
(b) The product of two integers having opposite signs is the product of their absolute values with a negative sign.

Division of integers
(a) If integers have the same sign, the quotient is always positive.
(b) If integers have opposite signs, the quotient will be negative.

Note:

· The integers are …, -3,-2,-1, 0,1, 2, 3,…

· 1, 2, 3, 4,… are called positive integers and -1,-2,-3,… are called negative integers. 0 is neither positive nor negative.

· Integer 0 is less than every positive integer but greater than every negative integer.

· The absolute value of an integer is the numerical value of the integer regardless of its sign.

· The absolute value of an integer is either positive or zero. It cannot be negative.

· The sum of two integers having the same sign is the sum of their absolute values with a positive sign.

· The sum of two integers having opposite signs is the difference of their absolute values with the sign of the greater absolute value.

· To subtract an integer b from a we change the sign of b and add, i.e., a + (-b)

· The product of two integers having the same sign is positive.

· The product of two integers having different signs is negative.

· Two integers, which when added give 0, are called additive inverse of each other.

· Additive inverse of zero is 0.

BASIC NUMBER PROPERTIES

The ideas behind the basic properties of real numbers are rather simple. You may even think of it as “common sense” math because no complex analysis is really required. There are four (4) basic properties of real numbers: namely; commutative, associative, distributive and identity. These properties only apply to the operations of addition and multiplication. That means subtraction and division do not have these properties built in.

I. COMMUTATIVE PROPERTY

For Addition

The sum of two or more real numbers is always the same regardless of the order in which they are added. In other words, real numbers can be added in any order because the sum remains the same.

Examples:

a) a + b = b + aa+b=b+a

b) 5 + 7 = 7 + 55+7=7+5

c) {}^ - 4 + 3 = 3 + {}^ - 4−4+3=3+−4

d) 1 + 2 + 3 = 3 + 2 + 11+2+3=3+2+1

For Multiplication

The product of two or more real numbers is not affected by the order in which they are being multiplied. In other words, real numbers can be multiplied in any order because the product remains the same.

Examples:

a) a \times b = b \times aa×b=b×a

b) 9 \times 2 = 2 \times 99×2=2×9

c) \left( { - 1} \right)\left( 5 \right) = \left( 5 \right)\left( { - 1} \right)(−1)(5)=(5)(−1)

d) m \times {}^ - 7 = {}^ - 7 \times mm×−7=−7×m

 

II. ASSOCIATIVE PROPERTY

For Addition

The sum of two or more real numbers is always the same regardless of how you group them. When you add real numbers, any change in their grouping does not affect the sum.

Examples:

 

For Multiplication

The product of two or more real numbers is always the same regardless of how you group them. When you multiply real numbers, any change in their grouping does not affect the product.

Examples:

 

 

III. IDENTITY PROPERTY

For Addition

Any real number added to zero (0) is equal to the number itself. Zero is the additive identity since a + 0 = aa+0=a or 0 + a = a0+a=a. You must show that it works both ways!

Examples:

 

For Multiplication

Any real number multiplied to one (1) is equal to the number itself. The number one is the multiplicative identity since a \times 1 = aa×1=a or 1 \times a = 11×a=1. You must show that it works both ways!

Examples:

 

IV. Distributive Property of Multiplication over Addition

Multiplication distributes over Addition

Multiplying a factor to a group of real numbers that are being added together is equal to the sum of the products of the factor and each addend in the parenthesis.

In other words, adding two or more real numbers and multiplying it to an outside number is the same as multiplying the outside number to every number inside the parenthesis, then adding their products.

Examples:

a)   

 

b)   

 

c)   

 

The following is the summary of the properties of real numbers discussed above:

 

Why Subtraction and Division are not Commutative

Maybe you have wondered why the operations of subtraction and division are not included in the discussion. The best way to explain this is to show some examples of why these two operations fail at meeting the requirements of being commutative.

If we assume that Commutative Property works with subtraction and division, that means that changing the order doesn’t affect the final outcome or result.

“Commutative Property for Subtraction”

Does the property a - b = b - aa−b=b−a hold?

a)   

 

b)   

 

Since we have different values when swapping numbers during subtraction, this implies that the commutative property doesn’t apply to subtraction.

“Commutative Property for Division”

Does the property a \div b = b \div aa÷b=b÷a hold ?

a)   

 

b)   

 

Just like in subtraction, changing the order of the numbers in division gives different answers. Therefore, the commutative property doesn’t apply to division.

Why Subtraction and Division are not Associative

If we want Associative Property to work with subtraction and division, changing the way on how we group the numbers should not affect the result.

Associative Property for Subtraction”

Does the problem \left( {a - b} \right) - c = a - \left( {b - c} \right)(a−b)−c=a−(b−c) hold?

a)   

 

b)   

 

These examples clearly show that changing the grouping of numbers in subtraction yield different answers. Thus, associativity is not a property of subtraction.

Associative Property for Division

Does the property \left( {a \div b} \right) \div c = a \div \left( {b \div c} \right)(a÷b)÷c=a÷(b÷c) hold?

a)   

 

I hope this single example seals the deal that changing how you group numbers when dividing indeed affect the outcome. Therefore, associativity is not a property of division.

UNIT 7

FRACTIONS

 

A fraction (from Latin fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction (examples: {\displaystyle {\tfrac {1}{2}}} and{\displaystyle {\tfrac {17}{3}}}) consists of a numerator displayed above a line (or before a slash), and a non-zero denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.

 

A cake with one quarter (one fourth) removed. The remaining three fourths are delimited by dotted lines and labeled by the fraction ¼

In positive common fractions, the numerator and denominator are natural numbers. The numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. The denominator cannot be zero, because zero parts can never make up a whole. For example, in the fraction 3⁄4, the numerator 3 tells us that the fraction represents 3 equal parts, and the denominator 4 tells us that 4 parts make up a whole. The picture to the right illustrates {\displaystyle {\tfrac {3}{4}}} or 3/4 of a cake.

A common fraction is a numeral which represents a rational number. That same number can also be represented as a decimal, a percent, or with a negative exponent. For example, 0.01, 1%, and 10⁻² are all equal to the fraction 1/100. An integer can be thought of as having an implicit denominator of one (for example, 7 equals 7/1).

Other uses for fractions are to represent ratios and division. Thus the fraction 3/4 can also be used to represent the ratio 3:4 (the ratio of the part to the whole), and the division 3 ÷ 4 (three divided by four). The non-zero denominator rule, which applies when representing a division as a fraction, is an example of the rule that division by zero is undefined.

We can also write negative fractions, which represent the opposite of a positive fraction. For example, if 1/2 represents a half dollar profit, then −1/2 represents a half dollar loss. Because of the rules of division of signed numbers (which states in part that negative divided by positive is negative), −1/2, −1/2 and 1/2 all represent the same fraction — negative one-half. And because a negative divided by a negative produces a positive, −1/2 represents positive one-half.

In mathematics the set of all numbers that can be expressed in the form a/b, where a and b are integers and b is not zero, is called the set of rational numbers and is represented by the symbol Q, which stands for quotient. A number is a rational number precisely when it can be written in that form (i.e., as a common fraction). However, the word fraction can also be used to describe mathematical expressions that are not rational numbers. Examples of these usages include algebraic fractions (quotients of algebraic expressions), and expressions that contain ir

FRACTION CONSTRUCTS

Understanding fractions means understanding all the possible concepts that fractions can represent. One of the commonly used meanings of fraction is part‐whole. But many who research fraction understanding believe students would understand fractions better with more emphasis across other meanings of fractions.

INTERPRETING FRACTIONS

Part- Whole: Using the part‐whole construct is an effective starting point for building meaning of fractions (Cramer & Whitney, 2010). Part‐whole can be shading a region, part of a group of people (35 of the class went on the field trip), or part of a length (we walked 312 miles).

Measurement:  Measurement involves identifying a length and then using that length as a measurement piece to determine the length of an object. For example, in the fraction 58, you can use the unit fraction 18 as the selected length and then count or measure to show that it takes five of those to reach 58. This concept focuses on how much rather than how many parts, which is the case in part‐whole situations (Behr, Lesh, Post, & Silver, 1983; Martinie, 2007).Division. Consider the idea of sharing $10 with 4 people. This is not a part‐whole scenario, but it still means that each person will receive one‐fourth (14) of the money, or 212 dollars. Division is often not connected to fractions, which is unfortunate. Students should understand and feel comfortable with the example here written as 104 , 4_10, 10 , 4, 224, and 212 (Flores,Samson, & Yanik, 2006).

Operator: Fractions can be used to indicate an operation, as in 45of 20 square feet or 23 of the audience was holding banners. These situations indicate a fraction of a whole number, and students may be able to use mental math to determine the answer. This construct is not emphasized enough in school curricula (Usiskin, 2007). Just knowing how to represent fractions doesn’t mean students will know how to operate with fractions, which occurs in various other areas in mathematics ( Johanning, 2008).

Ratio: Discussed at length in Chapter 18, the concept of ratio is yet another context in which fractions are used. For example, the fraction 14 can mean that the probability of an event is one in four. Ratios can be part‐part or part‐whole. For example, the ratio 34 could be the ratio of those wearing jackets (part) to those not wearing jackets (part), or it could be part‐whole, meaning those wearing jackets (part) to those in the class (whole).

 

Top of Form

FRACTION MODELS

The three major categories of fraction models are the area model, linear model, and set model.  Evidence suggests that providing opportunities for students to work with all three models plays a crucial role in developing a conceptual understanding of fractions. Having students repeat an activity with a different model and asking them to make connections between models can also be useful.  Too often, students learn rules for manipulating written fractions before they have developed an understanding of fraction concepts.  Using fraction models can help students clarify ideas that are often confused in a purely symbolic mode and construct mental referents that enable them to perform fraction tasks meaningfully.

THREE CATEGORIES OF FRACTION MODELS

1. AREA MODEL

In the area model fractions are represented as parts of an area or region. Useful manipulatives include rectangular or circular fraction sets, pattern blocks, geoboards and tangrams.

Rectangular, or circular fraction sets can be used to develop the understanding that fractions are parts of a whole, to compare fractions, to generate equivalent fractions and to explore operations with fractions.  While the rectangular model is easier for students to draw precisely, the circular model emphasizes the part-whole concept of fractions and the meaning of the relative size of a part to the whole (Cramer, Wyberg & Leavitt, 2008). Hence, students should have opportunities to work with both rectangular and circular models. 

When purchasing rectangular or circular fraction sets keep in mind that those with pieces that are not labelled provide more opportunities for learning. Labeled kits deprive students of opportunities to think about the size of the pieces in relation to the whole and also lead students to incorrectly assume that only one of the pieces can be considered as the whole, thereby making them less useful when working on concept of unit activities in which students name fractions when the unit is varied.
Sample Activities:
Make One Whole 
Equivalent Fractions Exploration
Adding Like Fractions
Subtracting Like Fractions

 

 

Pattern blocks allow students to see how shapes can be partitioned into other shapes. Consisting of blocks in six geometric shapes they are referred to as green triangles, orange squares, blue parallelograms, tan rhombuses, red trapezoids, and yellow hexagons.  When students work only with the yellow, red, blue, and green blocks, and the yellow block is given the value of 1, a red block represents one-half, a blue block represents one-third, and a green block represents one-sixth.
Sample Activities:
Pattern Block Fractions
Triangle Fractions
Quadrilateral Fractions
What Fraction is Red?

 

 

 

Geoboards can be used to explore a range of math concepts including properties of shapes, area, and perimeter.  Available in a variety of sizes they also provide an engaging way for students to partition wholes into equal areas and to explore equivalent fractions using an area model.
Sample Activities:
Partition a Square
Partition a Square
Partition a Square

 

The standard tangram consists of seven shapes that can be arranged to form a square. Frequently used for exploring geometry, they can also be used for problem solving tasks involving fractions. 
Sample Activity:
The Warlord's Puzzle

2. LINEAR MODEL

In the linear model lengths are compared instead of areas (e.g., 3/4 of an inch). Either number lines are drawn and subdivided or physical materials are compared on the basis of length.  Useful manipulatives include Cuisenaire rods or fraction strips that are easily connected to ideas about fractions on a number line.

Cuisenaire Rods are rectangular rods, each of a different color and size, that can be used to help students develop understanding of a range of math concepts including addition, subtraction, multiplication, length, area, volume, and fractions.  The lengths of the different colored rods increase incrementally from the smallest size (1cm long) to the largest size (10 cm long).  The rods are flexible because any length can be used to represent the whole. For example, in order to work with 1/4 and 1/8 students would use the brown rod which is 8 units long.  The purple rod would then become 1/2, the red rod would become 1/4, and the white rod would become 1/8.  
Sample Activities:
Cuisenaire Fractions
Representing Unit Fractions
Match the Length
Make One Whole Mixed Number x Fraction Models
Divide a Whole Number by a Unit Fraction

 

The number line is an important linear model for students to work with as it reinforces the fact that there is always one more fraction to be found between two fractions.
Sample Activities:
Fractions on a Number Line
Roll a Fraction

Fraction Strips can be used to explore equivalency, comparison of fractions, ordering fractions and number operations with fractions. If purchasing commercially made strips opt for those that are unlabeled to provide more opportunities for thinking and learning.
Sample Activities:
Making Fraction Strips

Making Fraction Strips
Equivalent Fractions Exploration

3. SET MODEL

In the set model the whole is understood to be a set of objects, and subsets of the whole make up fractional parts (e.g. ½ of the class, ¼ of a set of buttons, 1/3 of a tray of muffins).  Useful manipulatives include two-color counters, centimeter cubes, or any set of objects that can be counted (e.g. a bag of marbles).

Typically red on one side and yellow on the other two colour counters can be flipped to change their colour and model various fractional parts of a set.
Sample Activities:
Fractional Parts of a Set
Equivalent Fractions: Set Model

 

OTHER USEFUL MANIPULATIVES

Other useful manipulatives that can be used in math centers to develop fraction concepts and skills include dominoes, fraction cards and fraction dice.  When purchasing dominoes look for a Double Nine set (55 dominoes) so that you have a range of fractions greater than one .

Sample Activities:
Equivalent Fractions: Dominoes
Renaming Fractions Greater Than One

EQUIVALENT FRACTIONS 

Equivalent Fractions have the same value, even though they may look different.

These fractions are really the same:

1/2 = 2/4 = 4/8

Why are they the same? Because when you multiply or divide both the top and bottom by the same number, the fraction keeps it's value.

The rule to remember is:

"Change the bottom using multiply or divide,
And the same to the top must be applied"

Here is why those fractions are really the same:

And visually it looks like this:

Also see the Chart of Fractions with many examples of equivalent fractions.

DIVIDING

Here are some more equivalent fractions, this time by dividing:

Choose the number you divide by carefully so that the results (both top and bottom) stay whole numbers.

If we keep dividing until we can't go any further, then we have simplified the fraction (made it as simple as possible).

Summary:

· You can make equivalent fractions by multiplying or dividing both top and bottom by the same amount.

· You only multiply or divide, never add or subtract, to get an equivalent fraction.

· Only divide when the top and bottom stay as whole numbers.

TYPES OF FRACTIONS

Fractions have a numerator and a denominator

Fractions (and ratios) are made up of two numbers: the numerator and the denominator.

The denominator is the bottom number, and represents how many parts are possible—in other words, how many parts go into a whole.

The numerator is the top number, and represents how many parts you have. It might be less than the whole!

If the denominator is a 0 that is bad. We are not allowed to divide by zero.

If the denominator is a 1, our fraction is actually an integer. This suggests that we can add, subtract, multiply, or divide fractions. Adding and subtracting fractions is trickier, especially if we don't have common denominators. But it can be done.

For now, though we want to show you a visual of the different parts of a fraction.

 

Image Source: Zora Gilbert

Every fraction has a multiplicative inverse. The inverse of fraction ab is ba, and is also called the reciprocal. If we multiply a fraction by its multiplicative inverse, we get 1. The numerator and denominator cancel each other out.

UNIT 8

RATIONAL AND IRRATIONAL NUMBERS

 

RATIONAL NUMBERS

Example: Find three rational numbers between 3 and 5.

Solution:

PROPERTIES OF RATIONAL NUMBERS

Note: Since the sum of two rational numbers is always a rational number; we say the set of rational numbers is closed for addition.

In the same way the set of rational numbers is closed for:

· Subtraction

· Multiplication

· Division (if divisor not equal to zero)

DECIMAL REPRESENTATION OF RATIONAL NUMBERS

Now examine the following rational numbers:

These non-terminating decimals in which a digit or a set of digits repeats continually are called a recurring or a periodic or a circulating decimal. The repeating digit or the set of repeating digits is called the period of the recurring decimal.

Note: If the denominator of a rational number can be expressed as the power either of 2 or of 5 or of 2 and 5 both, the rational number is convertible into a terminating decimal. Otherwise, the rational number is convertible to a recurring decimal.

IRRATIONAL NUMBERS

Example: Find two irrational numbers between 2 and 3.

Solution:

If a and b are two positive numbers such that ab is not a perfect square then :

Example: Insert a rational and an irrational number between 3 and 4.

More about irrational numbers

1. The sum of two irrational numbers may or may not be irrational.

2. The difference of two irrational numbers may or may not be irrational.

3. The product of two irrational numbers may or may not be irrational.

4. The negative of an irrational number is always irrational.

5. The sum of a rational and an irrational number is always irrational.

6. The product of a non-zero rational number and an irrational number is always irrational.

PLACE VALUE

 

In math, every digit in a number has a place value. 

Place value can be defined as the value represented by a digit in a number on the basis of its position in the number. 

Here’s an example showing the relationship between the place or position and the place value of the digits in a number. 

In 13548, 1 is in ten thousands place and its place value is 10,000,

3 is in thousands place and its place value is 3,000,

5 is in hundreds place and its place value is 500,

4 is in tens place and its place value is 40,

8 is in ones place and its place value is 8.

 

Place Value Games

Understanding the place value of digits in numbers helps in writing numbers in their expanded form. For instance, the expanded form of the number above, 13548 is 10,000 + 3,000 + 500 + 40 + 8.

 

A place value chart can help us in finding and comparing the place value of the digits in numbers through millions. The place value of a digit increases by ten times as we move left on the place value chart and decreases by ten times as we move right. 

 

Here’s an example of how drawing the place value chart can help in finding the place value of a number in millions.

In 3287263, 3 is in millions place and its place value is 3000000,

2 is hundred thousands place and its place value is 200000,

8 is in ten thousands place and its place value is 80000,

7 is in thousands place and its place value is 7000,

2 is in hundreds place and its place value is 200,

6 is in ten place and its place value is 60,

3 is in ones place and its place value is 3.

 

The place value of digits in numbers can also be represented using base-ten blocks and can help us write numbers in their expanded form. 

Here’s how the number 13548 can be represented using base-ten blocks. 

 

Decimal Place Value

Decimal numbers are fractions or mixed numbers with denominators of powers of ten. In a decimal number, the digits to the left of the decimal point represent a whole number. The digits to the right of the decimal represent the parts. The place value of the digits becomes 10 times smaller.

The first digit on the right of the decimal point means tenths i.e.  .

 

In 27.356, 27 is the whole number part, 

2 is in tens place and its place value is 20,

7 is in ones place, and its place value is 7. 

There are three digits to the right of the decimal point,               

3 is in the tenths place, and its place value is 0.3 or,

5 is in the hundredths place, and its place value is 0.05 or,

6 is in the thousandths place, and its place value is 0.006 or.

 

 

GOOD LUCK WITH YOUR EXAMS...