# Comparing None Perfect Squares And Square Roots To Other Numbers Pdf

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## 5.2: Square Roots

Students should have a good understanding of what constitutes a rational number and should have experience working with irrational numbers to the extent that they understand they are non-terminating, non-repeating decimals that can be approximated from the following standards:. The teacher will activate prior knowledge by reviewing the relationship between the term "squaring a number" and finding the area of a square.

If a class set of square tiles are available for each student or pair of students to manipulate, that would be best. The teacher will ask the students to show one tile and say that each side length is 1 unit. Next, the students should use their square tiles to show a larger square that has a side length of 2 units. Students should show a 2 x 2 square. Students should be recognize that they are actually counting the number of squares that make up the larger square and not simply multiply the side lengths.

The students should show a square with side lengths of 3 units Should show a 3 x 3 square. The teacher asks, "What would be the area of the next largest square that we can make with whole number units? The teacher should create a list of the areas on the board: 1, 4, 9, They should recognize that these are perfect squares.

If not, the teacher will lead them to this realization. The teacher may also need to connect the relationship between finding the area of the squares and the term "square. The teacher should draw a number line on the board or use a piece of electrical tape run along the floor in a straight line following along the tile makes easy work of this.

There should be 5 or 10 equal intervals placed on the number line to show decimals between each whole number. The teacher should ask for a student to label the number line to show whole numbers and tenths.

The numbers represent the side lengths of each square. Below each whole number, a student should be called upon to place the area of the square with that side length. The students should understand that the numbers above the line represent the side length of each square, and the numbers below the line represent the areas of the squares with that side length.

The teacher asks, "Do you see any patterns in the numbers on the line? The teacher should ask the students to go back to working with the square tiles and create a square with an area of Students should recognize that they will not be able to form a square with whole tiles.

They will be able to create a rectangle, but a square cannot be constructed with whole number side lengths. Some students may recognize that they might be able to generate a square with fractional side lengths.

The teacher should proceed in asking the students to generate a square with an area of 6, 3, 12, or any area other than a perfect square. The areas requested should not necessarily be given in numerical order because the next question will lead them into the learning objective, "What do these areas have in common? What pattern do you see with the areas that are being asked? Here the teacher will introduce the radical sign, , which represents the square root.

The number inside the radical is called the radicand. The radical represents the operation of finding the positive real number that can be multiplied by itself to produce the radicand. It is essentially the inverse operation of "squaring" a number. If we wanted to know what number would be multiplied by itself to find the side length of a square with an area of 36 square units, we find the square root of 36,. The teacher should elicit from the students that even when areas are not whole numbers and the areas are not "perfect" in terms of finding a square with whole integer side lengths, there must be a number that we can square to find these areas.

These numbers are called irrational. They will be non-terminating and non-repeating numbers decimals that do not form a predictable pattern and never end. Go back to the number line and ask the students to predict where the side length of a square with an area of 10 might fall.

At minimum, they should recognize that it will fall between 3 and 4 since those side lengths produce areas of 9 and Elicit from students that 10 should be placed closer to 9 than to 16, since it is only one value greater than 9, and 7 values less than The teacher should ask the students, "If we want to get a more specific value for what number multiplied by itself would give the approximate product of 10, we can find the square of decimals.

We know that 10 is located between the perfect squares 9 and 16, which are squares of 3 and 4. So the square root of 10 would be located between 3 and 4. We see that when we rewrite our inequality, now we have 9. Our target number, 10, is 0. This tells us that an approximation of the square root of 10 is 3. The teacher should elicit from students how they might get an even more accurate approximation. Possible response: use a similar process to approximate to the hundredths place Classes that are more advanced can be asked to find the approximation to the nearest hundredth to practice Florida Standard for Mathematical Practice MAFS.

The teacher will assign partners. Then the teacher will display and ask students how they might begin to find the approximate square root of 3 using a number line. Possible response: Label 1 and 2 on the top and 1 and 4 directly underneath each, since 1 squared is 1 and 2 squared is 4; 3 would be between them.

Ask students for next step. Possible response: 1. Next, the teacher will display and ask students to draw a number line to find the approximate square root of 6. Based on the needs of the class, the teacher will provide only as much 'step-by-step' support as is needed to be successful. The teacher will ask students to share with a partner. The teacher should continue by displaying.

Students should work independently, share with their partner, and be prepared to present their findings. The teacher will circulate as the students work on their dry erase boards or paper. The teacher will look for correct procedures in using a number to approximate the side lengths. The teacher will close the lesson by asking students to provide a quick summary of how to find the rational approximation of an irrational number in a quick write.

The students should take no longer than 5 minutes to write a brief summary, jot down bullet points, or make a quick outline describing the following:. The teacher will collect the quick writes and review these steps to ensure all students leave with the same information. Students will take the Lesson Assessment see Attachments Section to demonstrate mastery of finding the approximate square root to the nearest tenth place.

They will show work and their thinking using a number line. Prior to the lesson, the teacher should assess students for prior knowledge by asking students to answer the following questions:. Students who are unable to correctly answer these questions will need extra support throughout the lesson.

The teacher will observe students' work during the lesson and diagnose and correct any misconceptions during the guided practice and independent practice phases. Throughout the lesson, the teacher will provide individual verbal feedback to students and provide leading questions when necessary to guide them to the correct conceptual understanding for approximating irrational numbers. Students could research the history of Pi and demonstrate how to get better approximations than 3.

Calculators are not recommended, as this lesson relies on students' ability to work with radicals in their simplest form without providing an irrational approximation. Fluency is one of the desired outcomes. This resource is likely to support student engagement in the following the Florida Standards for Mathematical Practices:. A verifications link was sent to your email at. Please check your spam folder. The website is not compatible for the version of the browser you are using.

Not all the functionality may be available. Please upgrade your browser to the latest version. Rate It! Resource Information. Aligned Standards. Students will use number lines to approximate square roots of non-perfect squares. General Information Subject s : Mathematics. Grade Level s : 8. Intended Audience: Educators. Instructional Time: 1 Hour s.

Resource supports reading in content area: Yes Freely Available: Yes. Keywords: irrational numbers, non-perfect square roots, square roots. Please log in using the Sign in section on top then try again. This resource requires special permission and only certain users have access to it at this time.

Attachments TI Independent Practice r. Students will: be able to compare the size of irrational numbers and locate them approximately on a number line. Prior Knowledge: What prior knowledge should students have for this lesson?

Students should have a good understanding of what constitutes a rational number and should have experience working with irrational numbers to the extent that they understand they are non-terminating, non-repeating decimals that can be approximated from the following standards: MAFS.

Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

Guiding Questions: What are the guiding questions for this lesson? How can we find the square root of a "non-perfect" square? What does it mean to "square" a number? Why are square roots of "non-perfect" squares irrational?

## estimating non perfect squares

Skip to main content. Search form Search. Index notation worksheet year 7. Index notation worksheet year 7 index notation worksheet year 7 Treat it like a race to see who can complete the most and they will sit and concentrate for ages. Capacity Worksheets A great worksheet to help introduce the use of letters in maths. Transformations of Graphs. Writing your answers in index form, calculate: a 10

In mathematics , the irrational numbers are all the real numbers which are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable , meaning that they share no "measure" in common, that is, there is no length "the measure" , no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Like all real numbers, irrational numbers can be expressed in positional notation , notably as a decimal number. In the case of irrational numbers, the decimal expansion does not terminate, nor end with a repeating sequence.

Options include PDF or html worksheet, radicand range, perfect squares only, font size, workspace, and more. If you choose to allow non-perfect squares, the answer is typically an unending decimal that is rounded to a certain number of digits. Square roots and one other operation; round answers to 3 decimals (grades.

## Approximating square roots

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Students should have a good understanding of what constitutes a rational number and should have experience working with irrational numbers to the extent that they understand they are non-terminating, non-repeating decimals that can be approximated from the following standards:. The teacher will activate prior knowledge by reviewing the relationship between the term "squaring a number" and finding the area of a square. If a class set of square tiles are available for each student or pair of students to manipulate, that would be best. The teacher will ask the students to show one tile and say that each side length is 1 unit. Next, the students should use their square tiles to show a larger square that has a side length of 2 units.

*When solving a problem with square roots , life will not always give you perfect squares. We suggest that you memorize the first 12 perfect squares. For the most part we won't work with numbers bigger than *

### Testing Imperfection

By that I mean, how did an irrational number come into existence. IRRational vs. Given problem situations that include pictorial representations of irrational numbers, the student will find the approximate value of the irrational numbers. Estimating Square Roots We find the two perfect squares that are before and after the square root of 8. Find lesson plan resources, sample questions, apps and videos for 8 grade lesson - Approximating Irrational Numbers. Interactive video lesson plan for: Math 8 Lesson 3: Estimating Irrational Numbers Simplifying Math Activity overview: In this lesson we will see how to estimate irrational square roots without using a calculator.

As students arrive in class, I give each group the square tiles used on Day 1. Once all of the groups are seated I will ask students to work together to represent each of the following numerical expressions using the tiles:. As students work on this task, I will walk about the room taking pictures of different student work samples.

If you're seeing this message, it means we're having trouble loading external resources on our website. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. Donate Login Sign up Search for courses, skills, and videos. Math 8th grade Numbers and operations Approximating irrational numbers. Approximating square roots. Approximating square roots walk through. Practice: Approximating square roots.

compare real numbers. Square roots of numbers that are not perfect In other words, is the same as. square roots of perfect squares to help estimate the.

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