A decimal is a way of representing a part of a whole number or a fraction. It is based on the powers of 10, with each place value to the right of the decimal point representing a power of 10 that is a negative exponent. For example, in the number 123.456, the digit 1 is in the hundreds place, the digit 2 is in the tens place, the digit 3 is in the ones place, the digit 4 is in the tenths place, the digit 5 is in the hundredths place, and the digit 6 is in the thousandths place.

Decimals are written using a decimal point to separate the whole number part from the fractional part. For example, the number 3.14 represents 3 whole units and 14 hundredths. The digits to the right of the decimal point represent the fractional part of the number.

When comparing decimals, start from the left and compare the digits in each place value. If the digits are the same, move to the next place value. If the digits are different, the larger digit indicates the larger number. For example, when comparing 0.35 and 0.4, start with the tenths place: 3 is less than 4, so 0.35 is less than 0.4.

When adding or subtracting decimals, align the decimal points and perform the arithmetic as if the decimals were whole numbers. Then, place the decimal point in the answer directly below the decimal points in the numbers being added or subtracted.

When multiplying decimals, ignore the decimal points and multiply the numbers as if they were whole numbers. Then, count the total number of decimal places in the original numbers and place the decimal point in the answer so that it has the same number of decimal places. When dividing decimals, move the decimal point to the right in both the dividend and divisor until the divisor is a whole number, then perform the division as if it were whole numbers. Finally, place the decimal point in the quotient directly above the decimal point in the dividend.

To round a decimal number, find the place value you want to round to and look at the digit to the right of that place value. If the digit is 5 or greater, round up; if it is less than 5, round down. Then, replace all the digits to the right of the chosen place value with zeros.

- Understand the concept of decimal notation and the place value system.
- Practice comparing decimals and understanding their relative magnitudes.
- Learn the rules for adding, subtracting, multiplying, and dividing decimals.
- Practice rounding decimals to a given place value.
- Apply decimal operations to real-life situations, such as shopping or calculating measurements.

Study GuideSimilarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer key

Similarity and scale Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons Worksheet/Answer keyUsing Similar Polygons Worksheet/Answer keySimilar Polygons

Number and Operations (NCTM)

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Understand and use ratios and proportions to represent quantitative relationships.

Compute fluently and make reasonable estimates.

Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.

Geometry (NCTM)

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.

Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects.

Create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

Apply transformations and use symmetry to analyze mathematical situations.

Describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling.

Measurement (NCTM)

Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, using ratio and proportion.

Grade 8 Curriculum Focal Points (NCTM)

Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle

Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean Theorem is valid by using a variety of methods - for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.