All grade levels include children's literature recommendations and most include one or more titles. Find out more. An independent study found that students who use Bridges perform better than their peers. Read full report. Bridges draws upon decades of research into the best methods for teaching and learning math. View research base. This two-day summer workshop brings our workshop leaders to your district. They work face-to-face with your classroom teachers, walking through unpacking of the curriculum components and exploring math practices in the contexts of the models and strategies. Each workshop is packed with information, pedagogy, and hands-on activities.

Teachers explore resources, dive into Number Corner, and do math that resembles a Bridges classroom. Bridges Getting Started workshops are scheduled either by grade-level bands, such as K—2 and 3—5, or are grade specific. A maximum of 30 participants is allowed in each session, but multiple sessions can be scheduled. Thank you for sharing your expertise, bringing laughter to the room, and cultivating confidence for our teachers. Learn More. This three-day workshop is designed to equip instructional leaders to lead their own Bridges Getting Started Workshops in-district for new hires or new-to-grade-level teachers.

Participants will experience a condensed Bridges Getting Started Workshop covering both K—2 and 3—5 grade bands. They'll leave with the knowledge and materials necessary to offer these workshops in their own district. This face-to-face workshop is appropriate for districts after their first year of implementing Bridges in Mathematics. This week online course is offered in both fall and winter.

It is designed to support math coaches and instructional leaders working towards increasing student achievement and teacher content knowledge during a Bridges in Mathematics implementation. Participants read, reflect, and respond to prompts every two weeks. The readings were helpful because I could directly apply them to the work I have been doing. Prior to the workshop, we work with you to determine these needs.

Our workshop leaders come to your district and work face-to-face with your classroom teachers. These leaders offer guidance in specific areas where teachers struggle and model methods for successful implementation. We appreciated having an expert on-site to answer our questions, share resources, and share their experience. The annual Leadership Institute offers lead teachers, coaches, and curriculum specialists the tools for developing and sustaining a successful implementation of Bridges in Mathematics, Number Corner, and Bridges Intervention. This institute is limited to districts after their first year of implementing Bridges in Mathematics 2nd Edition.

I left feeling enthusiastic and excited as we head into our second year of using Bridges. Bridges has been so effective. My students are challenged and engaged, and parents and administrators are very impressed. I am very confident that my students learned far more in math this year than in previous years.

Every kid in America deserves access to this quality of a universal core resource. One student wanted to cancel spring break so we could keep doing math. I was astounded. Bridges really helped me pull all of the pieces together in terms of content, delivery, and motivation. My students really excel in math, and I owe much of that to this curriculum. My daughter never liked math before this year. Now she will not miss school for anything because she'd miss Bridges. That math program you're using is great!

I can say without reservation that Bridges is one of the best decisions made in my 14 years here. Bridges is the best program I ever had the privilege of being a part of in my 30 year teaching career. Students who have used the Bridges program in earlier years bring prior knowledge of many math concepts which makes a real difference in the pace that I can achieve. Many students, especially my ELL students, are developing a stronger sense of number using the visual ten-frames and finger models from Bridges. These strategies are providing successful concrete to representational forms for students.

I wanted to let you know how much my students and I enjoyed the "quick sketch" model for multiplying larger numbers. It is nice to be able to put a visual in student's hands to help them understand a normally complicated process. Bridges promotes a deeper level of understanding and encourages higher order thinking. Bridges met the needs of both my high and low students. The kids were challenged and really enjoyed math -- even those who started the year hating math. And, my students did well on the state math assessment.

It seems everyday one of our teachers stops me in the hall to share what great things are happening in math. This program has given me the steps to make that dream a reality.

## Download e-book Grade 6 Mathematics Lesson 3 (iPad Math Learning Guides)

For the first time in my 11 years of teaching I feel my class truly has a good grasp of fractions. Great job getting my kids there! The children can visualize math and understand patterns in it. They are much more capable of exploring how they got an answer. All of the children enjoy math and are actively involved. I have a very challenging group of kids, and we are having an outstanding year in math. The kids are so engaged and interested.

In more than 20 years of teaching elementary math I have used a range of curricula in several states and two countries. The visual models in Bridges are one of its huge strengths! They help all children understand the math concepts— students who might otherwise struggle with the content, and those ready for the challenge of generalizing the math concepts to larger numbers and different situations.

That tells it all! Our current fifth graders have been using Bridges since kindergarten. Their teachers are extremely impressed with the math vocabulary and deep understanding these students bring forth in the classroom. I used to wish math would go by fast because I didn't get it and I also hoped my teacher wouldn't call on me. Can you represent them on a number line? Recall what rational numbers are Why is that? Neither can be expressed as a quotient of two integers. We can locate rational numbers on the real number line.

Get the midpoint of the segment from 0 to 1. Locate 1. The number 1. Divide the segment from 0 to 2 into 8 equal parts. The 7th mark from 0 is the point 1. Example 3. Locate the point on the number line. Dividing the segment from -2 to 0 into 6 equal parts, it is easy to plot. The number is the 5th mark from 0 to the left. Go back to the opening activity. You were asked to locate the rational numbers and plot them on the real number line.

Before doing that, it is useful to arrange them in order from least to greatest. To do this, express all numbers in the same form — either as similar fractions or as decimals. Because integers are easy to locate, they need not take any other form. Therefore, plotting them by approximating their location gives III. Locate and plot the following on a number line use only one number line. Name 10 rational numbers that are greater than -1 but less than 1 and arrange them from least to greatest on the real number line?

Name one rational number x that satisfies the descriptions below: a. By changing all rational numbers to equivalent forms, it is easy to arrange them in order, from least to greatest or vice versa. In this lesson, you will learn techniques in adding and subtracting rational numbers. Techniques include changing rational numbers into various forms convenient for the operation as well as estimation and computation techniques.

Express rational numbers from fraction form to decimal form terminating and repeating and non-terminating and vice versa; 2. Add and subtract rational numbers; 3. Solve problems involving addition and subtraction of rational numbers. Forms of Rational Numbers I. Activity 1. Change the following rational numbers in fraction form or mixed number form to decimal form: a. Change the following rational numbers in decimal form to fraction form.

Discussion Non-decimal Fractions There is no doubt that most of the above exercises were easy for you. 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.

## Understanding and Representing Ratios

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?

### NCERT Solutions for Class 6 Maths Chapter 9 - Data Handling

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. 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. 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. 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.

## Stolen Child

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. 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?

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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? 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. 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.