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Key characteristics of the curriculum

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Best Math Apps for High School

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Illustrative Mathematics Grade 6 - Unit 1- Lesson 3

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These qualities will help you to ensure that you choose apps that will stimulate, challenge and inspire your students and enhance their overall learning experience. Jul 16, Educational App Store. This is because all except item 2f are what we call decimal fractions. These numbers are all Likewise, the number What do you do when the rational number is not a decimal fraction? How do you convert from one form to the other?

Remember that a rational number is a quotient of 2 integers. To change a rational number in fraction form, you need only to divide the numerator by the denominator. The smallest power of 10 that is divisible by 8 is The smallest power of 10 that is divisible by 16 is 10, To change rational numbers in decimal forms, express the decimal part of the numbers as a fractional part of a power of For example, What about non-terminating but repeating decimal forms?

How can they be changed to fraction form? Study the following examples: Activity Recall that we added and subtracted whole numbers by using the number line or by using objects in a set. Using linear or area models, find the sum or difference. Consider the following examples: 1. Since there is only 1 repeated digit, multiply the first equation by Since there are 2 repeated digits, multiply the first equation by Answer the following questions: 1. Is the common denominator always the same as one of the denominators of the given fractions?

Is the common denominator always the greater of the two denominators? What is the least common denominator of the fractions in each example? Is the resulting sum or difference the same when a pair of dissimilar fractions is replaced by any pair of similar fractions? Problem: Copy and complete the fraction magic square.

The sum in each row, column, and diagonal must be 2. Examples: To Add: To Subtract: a. You were asked to find the sum or difference of the given fractions. You would have to apply the rule for adding or subtracting similar fractions. Not always.

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Their least common denominator is 20 not 5 or 4. The least common denominator is always greater than or equal to one of the two denominators and it may not be the greater of the two denominators. Is the resulting sum or difference the same as when a pair of dissimilar fractions is replaced by any pair of similar fractions? Yes, for as long as the replacement fractions are equivalent to the original fractions. Perform the indicated operations and express your answer in simplest form. Give the number asked for.

What is three more than three and one-fourth? Subtract from the sum of. What is the result? Increase the sum of. What is? Solve each problem. Michelle and Corazon are comparing their heights. What is the difference in their heights? Angel bought meters of silk, meters of satin and meters of velvet.

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How many meters of cloth did she buy? Arah needs kg. Tan has liters of gasoline in his car. He wants to travel far so he added 16 liters more. How many liters of gasoline is in the tank? After boiling, the liters of water was reduced to 9 liters. How much water has evaporated? Express the decimal numbers in fractions then add or subtract as described earlier. Example: Add: 2. Arrange the decimal numbers in a column such that the decimal points are aligned, then add or subtract as with whole numbers. Perform the indicated operation. Solve the following problems: a.

Helen had P for shopping money. When she got home, she had P How much did she spend for shopping? Ken contributed P How much were they able to gather altogether? If I subtract If I increase my number by Kim ran the meter race in Tyron ran faster by SUMMARY This lesson began with some activities and instruction on how to change rational numbers from one form to another and proceeded to discuss addition and subtraction of rational numbers.

The exercises given were not purely computational. While there are rules and algorithms to remember, this lesson also shows why those rules and algorithms work. Multiply rational numbers; 2. Divide rational numbers; 3.

UnboundEd Mathematics Guide

Solve problems involving multiplication and division of rational numbers. Lesson Proper A. Models for the Multiplication and Division I. Activity: Make a model or a drawing to show the following: 1. A pizza is divided into 10 equal slices. Kim ate of of the pizza. What part of the whole pizza did Kim eat? Miriam made 8 chicken sandwiches for some street children. She cut up each sandwich into 4 triangular pieces. If a child can only take a piece, how many children can she feed? Can you make a model or a drawing to help you solve these problems?

A model that we can use to illustrate multiplication and division of rational numbers is the area model. Suppose we have one bar of chocolate represent 1 unit. Divide the bar first into 4 equal parts vertically. What about a model for division of rational numbers? One unit is divided into 5 equal parts and 4 of them are shaded. Each of the 4 parts now will be cut up in halves Since there are 2 divisions per part i.

How then can you multiply or divide rational numbers without using models or drawings? Important Rules to Remember The following are rules that you must remember. To multiply rational numbers in fraction form simply multiply the numerators and multiply the denominators. To divide rational numbers in fraction form, you take the reciprocal of the second fraction called the divisor and multiply it by the first fraction. In symbol, where b, c, and d are NOT equal to zero. Example: Multiply the following and write your answer in simplest form a. The easiest way to solve for this number is to change mixed numbers to an improper fraction and then multiply it.

Or use prime factors or the greatest common factor, as part of the multiplication process. Write your answer on the spaces provided: 1. Find the products: a.

Contexts for Learning Mathematics

Divide: 1. Solve the following: 1. Julie spent hours doing her assignment. Ken did his assignment for times as many hours as Julie did. How many hours did Ken spend doing his assignment? How many thirds are there in six-fifths? Hanna donated of her monthly allowance to the Iligan survivors. If her monthly allowance is P, how much did she donate? The enrolment for this school year is If are sophomores and are seniors, how many are freshmen and juniors?

The four employees each took home the same amount of leftover cake.

How much did each employee take home? Take the reciprocal of , which is then multiply it with the first fraction. Using prime factors, it is easy to see that 2 can be factored out of the numerator then cancelled out with the denominator, leaving 4 and 3 as the remaining factors in the numerator and 11 as the remaining factors in the denominator. Multiplication and Division of Rational Numbers in Decimal Form This unit will draw upon your previous knowledge of multiplication and division of whole numbers.

Recall the strategies that you learned and developed when working with whole numbers. Activity: 1. Give students several examples of multiplication sentences with the answers given. Place the decimal point in an incorrect spot and ask students to explain why the decimal place does not go there and explain where it should go and why. Example: Five students ordered buko pie and the total cost was P How much did each student have to pay if they shared the cost equally?

Questions and Points to Ponder: 1. In multiplying rational numbers in decimal form, note the importance of knowing where to place the decimal point in a product of two decimal numbers. Do you notice a pattern? In dividing rational numbers in decimal form, how do you determine where to place the decimal point in the quotient?

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Arrange the numbers in a vertical column. Multiply the numbers, as if you are multiplying whole numbers. Starting from the rightmost end of the product, move the decimal point to the left the same number of places as the sum of the decimal places in the multiplicand and the multiplier. If the divisor is a whole number, divide the dividend by the divisor applying the rules of a whole number. The position of the decimal point is the same as that in the dividend. If the divisor is not a whole number, make the divisor a whole number by moving the decimal point in the divisor to the rightmost end, making the number seem like a whole number.

Move the decimal point in the dividend to the right the same number of places as the decimal point was moved to make the divisor a whole number. Lastly divide the new dividend by the new divisor. Perform the indicated operation 1. Finds the numbers that when multiplied give the products shown.

You also learned the rules for multiplying and dividing rational numbers in both the fraction and decimal forms. You solved problems involving multiplication and division of rational numbers. Objectives: In this lesson, you are expected to 1. Describe and illustrate the different properties of the operations on rational numbers. Apply the properties in performing operations on rational numbers. What is the missing number in item 1? How do you compare the answers in items 1 and 2 3. What about item 3? What is the missing number? In item 4, what number did you multiply with 1 to get? What number should be added to in item 5 to get the same number?

What is the missing number in items 6 and 7? What can you say about the grouping in items 6 and 7? What do you think are the answers in items 8 and 9? What operation did you apply in item 10? Problem: Consider the given expressions: a. If yes, state the property illustrated. For example: a. If are any rational numbers, then For example: 5. If are any rational numbers, then For example: 6.

For example: 7. For example: Question to Ponder Post-Activity Discussion Let us answer the questions posed in the opening activity. What is the missing number in item1? How do you compare the answers in items 1 and 2? When you multiply a number with zero the product is zero. What do you think is the missing number in items 6 and 7?

Exercises: Do the following exercises. Write your answer in the spaces provided. State the property that justifies each of the following statements. Find the value of N in each expression 1. The properties are useful because they simplify computations on rational numbers. These properties are true under the operations addition and multiplication.

Note that for the Distributive Property of Multiplication over Subtraction, subtraction is considered part of addition. Think of subtraction as the addition of a negative rational number. The key is to introduce them by citing useful examples. Activities A. Take a look at the unusual wristwatch and answer the questions below. Can you tell the time? What time is shown in the wristwatch? How will you describe the result?

What value could you get? Taking the square root of a number is like doing the reverse operation of squaring a number. Integers such as 1, 4, 9, 16, 25 and 36 are called perfect squares. Rational numbers such as 0. Perfect squares are numbers that have rational numbers as square roots. The square roots of perfect squares are rational numbers while the square roots of numbers that are not perfect squares are irrational numbers. Any number that cannot be expressed as a quotient of two integers is an irrational number. Decimal numbers that are non-repeating and non-terminating are irrational numbers.

Yes 2. What time is it in the wristwatch? They are all positive integers. Let us give the values asked for in Activity B. Using a scientific calculator, you probably obtained the following: 1. This means that -7 is a 2nd or square root of 49, 2 is a 4th root of 16 and is a 3rd or cube root of However, we are not simply interested in any nth root of a number; we are more concerned about the principal nth root of a number.

The principal nth root of a positive number is the positive nth root. The principal nth root of a negative number is the negative nth root if n is odd. If n is even and the number is negative, the principal nth root is not defined. In this expression, n is the index and b is the radicand. The nth roots are also called radicals. If it is, then the root is rational. Otherwise, it is irrational. Problem 1.

Tell whether the principal root of each number is rational or irrational. Estimating is very important for all principal roots that are not roots of perfect nth powers. Problem 2. The principal roots below are between two integers. Find the two closest such integers. Problem 3. Estimate each square root to the nearest tenth.

Now, take the square of 6. Since Now, compute for the squares of numbers between 6 and 6. Since 40 is close to Now take the square of 3. Compute for the squares of numbers between 3 and 3. Since 12 is closer to The square of Since is closer to Problem 4. Locate and plot each square root on a number line. Tell whether the principal roots of each number is rational or irrational.

Between which two consecutive integers does the square root lie? Estimate each square root to the nearest tenth and plot on a number line. Which point on the number line below corresponds to which square root? You learned to find two consecutive integers between which an irrational square root lies. You also learned how to estimate the square roots of numbers to the nearest tenth and how to plot the estimated square roots on a number line.

Intuitively, the absolute value of a number may be thought of as the non-negative value of a number. The concept of absolute value is important to designate the magnitude of a measure such as the temperature dropped by 23 the absolute value degrees. A similar concept is applied to profit vs loss, income against expense, and so on. Objectives: In this lesson, you are expected to describe and illustrate a. How far would the North Avenue station be from Taft Avenue? How far would she have travelled? How far would each have travelled from the starting point to their destinations?

What can you say about the directions and the distances travelled by Archie and Angelica? Questions To Ponder: 1. What subsets of real numbers are used in the problem? Represent the trip of Archie and Angelica to the house of Aloys using a number line. What are opposite numbers on the number line? Give examples and show on the number line.

What does it mean for the same distance travelled but in opposite directions? How can we represent the absolute value of a number? What notation can we use? Absolute Value — of a number is the distance between that number and zero on the number line. Number Line —is best described as a straight line which is extended in both directions as illustrated by arrowheads.

A number line consists of three elements: a. Let's look at the number line: The absolute value of a number, denoted " " is the distance of the number from zero. This is why the absolute value of a number is never negative. In thinking about the absolute value of a number, one only asks "how far? Warning: The absolute-value notation is bars, not parentheses or brackets. Use the proper notation; the other notations do not mean the same thing. It is important to note that the absolute value bars do NOT work in the same way as do parentheses. Solution: Given — — 3 , first find the absolute value of — 3.

Travelling 3 miles west can be represented by -3 pronounced negative 3. Two integers that are the same distance from zero in opposite directions are called opposites. The absolute value of a number is its distance from zero on the number line. Opposite numbers have the same absolute values. Realize significant savings over self-printing: We provide high-quality student workbooks delivered directly to your district or school. Each consumable student workbook covers three units within a grade level.

Sample Grade 6 Unit 1 and Unit 9. Sample Grade 7 Unit 1 and Unit 9. Sample Grade 8 Unit 1 and Unit 9. We work closely with districts to support the effective implementation of the LearnZillion Illustrative Mathematics 6—8 Math curriculum. The year-long engagement provides teachers, leaders, and coaches on-site, virtual, and on-demand support.

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