MANIPULATIVE MATHEMATICS FOR STUDENTS

Transcription

1 MANIPULATIVE MATHEMATICS FOR STUDENTS

2 Manipulative Mathematics Using Manipulatives to Promote Understanding of Elementary Algebra Concepts Lynn Marecek MaryAnne Anthony-Smith This file is copyright 07, Rice University. All Rights Reserved.

3 Table of Contents To the Student Page Manipulative Activities Page Game of The Number Line Part -Counting Numbers and Whole Numbers 8 The Number Line Part -Integers 0 The Number Line Part -Fractions Multiplication/Factors 7 Square Numbers Model Fractions Fractions Equivalent to One 6 Mixed Numbers and Improper Fractions 8 Equivalent Fractions Model Fraction Multiplication 7 Model Fraction Division Model Fraction Addition 6 Model Fraction Subtraction 50 Model Finding the Least Common Denominator 5 Addition of Signed Numbers 59 Subtraction of Signed Numbers 6 Multiples 65 Prime Numbers 68 Subtraction Property of Equality 70 Division Property of Equality 77 Visualizing Area and Perimeter 8 Measuring Area and Perimeter 87 Coin Lab 96 Exploring Slopes of Lines 99 Slope of Line Between Two Points 0 This file is copyright 07, Rice University. All Rights Reserved.

4 To the Student: Research has shown that our brains learn best when we start with concrete objects and then move on to abstract ideas. Manipulatives are concrete objects used to model abstract mathematical concepts. Although the word manipulatives may be new to you, using manipulatives is probably something you have already done. If you learned to add numbers by counting your fingers, you were using manipulatives. And maybe you used two-color counters or fraction tiles in a previous math class. You will see Illustrations of manipulatives in Elementary Algebra whenever a new concept is introduced. Then you will be encouraged to do a Manipulative Mathematics activity to help you develop a solid understanding of the concept. This booklet contains the Manipulative Mathematics activities that accompany Elementary Algebra. Each activity relates to a mathematical topic covered in Elementary Algebra. There are two parts to each activity: The student worksheet will lead you through a series of questions to help you understand a mathematical property or procedure through the use of manipulatives. The extra practice problems reinforce the results of the student worksheet. Most Extra Practice worksheets include a link to a website with online (or virtual ) manipulatives. Ask your math instructor if your college has manipulatives for you to use. You can access the virtual manipulatives online any time, /7. We have sets of manipulatives, such as color chips, fraction pieces, algebra tiles, geoboards, and more, in our classrooms. Our college students use them to model and learn about critical mathematical concepts and procedures. We have heard many a-ha s from students who finally understand, for example, how to work with fractions or signed numbers. We sincerely hope you will use our Manipulative Mathematics activities to help you succeed in Elementary Algebra. We wholeheartedly believe that through the use of manipulatives, you can develop an understanding of mathematics that translates into success throughout your mathematics courses. Lynn Marecek MaryAnne Anthony-Smith This file is copyright 07, Rice University. All Rights Reserved.

5 Manipulative Mathematics Game of Name The Game of Twenty-four is a great way to think mathematically. Given four numbers, you add, subtract, multiply and/or divide them so that the result is. You must use each number once--but only once. Start with the numbers,,, and 8. ) How can you use these numbers to create? Don t worry yet about,,, and 8. Think of pairs of any two numbers that multiply to. List some of the pairs here: ) First, let s think of as the product of 8. We want to combine,,, 8 to get and 8. (a) One way is to use minus to get, then times 8 is. But we need to use the number. How can we use the and still have? times is still. Putting this all these steps together using good algebra notation gives (8)(). Verify that this expression simplifies to. 8 (b) Here is another way to use the same four numbers,,,, 8, to get the product 8 : times is, and then minus gives. Finally multiply that by 8 to get. Show that this expression simplifies to : 8 ) This time, we ll use the fact that is the product of 6. (a) Can we combine,,, 8 to make 6 times? Well, plus is, 8 minus gives 6, and then 6 times is. Show that this expression simplifies to : 8 (b) Can you think of another combination? Using good algebra notation, write a different expression and show that it simplifies to. This file is copyright 07, Rice University. All Rights Reserved.

6 ) Another number fact that might help make is. (a) How can you combine,,, 8 to create and? plus 8 is, and plus is. Then twelve times two is! Write this as one expression using good algebra notation, then show that it simplifies to. (b) Can you think of another combination? Using good algebra notation, write a different expression and show that it simplifies to. Now use the numbers 5,, 5, to make. 5) Verify that each expression simplifies to. (a) 5 5 (b) 5 5 6) Using good algebra notation, write a different expression that simplifies to. Next try, 6, 6, 9. 7) Verify that each expression simplifies to. (a) (b) ) Using good algebra notation, write a different expression that simplifies to. 5 This file is copyright 07, Rice University. All Rights Reserved.

7 Manipulative Mathematics Game of Extra Practice Name For each set of numbers use good algebra notation to write different expressions that simplify to. ),,, (a) (b) ),, 5, 9 (a) (b) ),, 7, 8 (a) (b) ), 7, 8, 9 (a) (b) 5),, 6, 6 (a) (b) 6),,, 6 (a) (b) 6 This file is copyright 07, Rice University. All Rights Reserved.

8 7),,, 5 (a) (b) 8),,, 5 (a) (b) 9),, 5, 7 (a) (b) 0),, 7, 9 (a) (b) For more practice, there are several websites where you can play the Game of Twenty-four online. One of them is 7 This file is copyright 07, Rice University. All Rights Reserved.

9 Manipulative Mathematics Number Line Name The Number Line Part -- Counting Numbers and Whole Numbers Counting numbers and whole numbers can be visualized by creating a number line. ) To create your own number line: (a) Take a strip of paper about feet long and fold it lengthwise to make a straight crease. (b) Open the fold and draw a line in the crease. Put an arrow at each end of the line to indicate that the line continues. (c) Mark a point at about the middle of the line. Label that point 0. This point is called the origin. 0 ) Choose a convenient unit and mark off several of these units to the right of 0. Pair these points with the numbers,,,, 5, and so on. When a number is paired with a point, we call it the coordinate of the point ) Draw a red triangle around each counting number. ) Draw a blue circle around each whole number. 5) Notice that all the numbers on your number line except 0 are marked with both a triangle and a square. What conclusion can you draw from this? 6) In one corner of your strip make a key that explains the symbols around the numbers. Counting numbers Whole numbers 7) Put your number line in your notebook for future use, so you can add more numbers to the number line as you proceed through this course. 8 This file is copyright 07, Rice University. All Rights Reserved.

10 Manipulative Mathematics Number Line Part I Extra Practice Name Name the coordinate of each point. ) 7 ) ) Locate each point on the number line. ) 5 5) 5 6 6) For each set of numbers identify (a) the counting numbers and (b) the whole numbers. 7) 0,,, 7.5,, 99 5 (a) (b) 8) 9) 0,,, 5, 6, 99.9, 50 0,,., 6, 0, 88,.5 9 (a) (a) (b) (b) 0) 5 0,,, 5., 8,.99, 65, 00 (a) (b) 9 This file is copyright 07, Rice University. All Rights Reserved.

11 Manipulative Mathematics Number Line Name The Number Line Part -- Integers The number line you made in Part started at 0. All the numbers you have worked with so far have been positive numbers, numbers greater than Now you need to expand your number line to include negative numbers, too. Negative numbers are numbers less than zero. So the negative numbers will be to the left of zero on the number line. Get your number line out of your notebook and place it on your desk. ) Mark off several units to the left of zero. Make sure your unit is the same size as the one you used on the positive side. ) Now label at the first point left of 0, then at the next point to the left, and so on. 0 negative numbers zero positive numbers ) The arrows on both ends of the number line indicate that the numbers keep going forever. (a) Is there a largest positive number? (b) Is there a smallest negative number? ) Is zero a positive or a negative number? Numbers larger than zero are positive and numbers smaller than zero are negative. Zero is neither positive nor negative. 5) Locate and label the following points on this number line. (a) (b) (c) (d) 5 (e) The whole numbers and their opposites are called the integers. Integers:...,,, 0,,,,... 6) Put a black square around each integer on your number line. 7) What do you notice about the integers, counting numbers and whole numbers on your number line? 0 This file is copyright 07, Rice University. All Rights Reserved.

12 Manipulative Mathematics Number Line Part Extra Practice Name Name the coordinate of each point. ) 6 0 ) 8 ) Locate each point on the number line. ) ) 9 8 6) For each set of numbers identify the (a) counting numbers, (b) whole numbers, and (c) integers. 7) 9,, 0,, 5, 7.5, 0 (a) (b) (c) 8),, 0,,.65, 9, 8 (a) (b) (c) 6 9) ,,, 0,,, 99 9 (a) (b) (c) 0) 5,.5,, 0,, 0, 8.(a) (b) (c) 7 This file is copyright 07, Rice University. All Rights Reserved.

13 Manipulative Mathematics Number Line Name The Number Line Part -- Fractions Now you are ready to include fractions on your number line. This will help you visualize fractions and understand their value. Take your number line out of your notebook and place it on your desk. Our goal is to locate the numbers 7 9 8,,,,,, 5, and 5 5 on the number line. ) We ll start with the whole numbers and 5 because they are the easiest to plot. Put points to mark and 5. 5 ) The proper fractions listed are and 5 5. (a) Proper fractions have value less than one. Between which two whole numbers are the proper fractions and located? They are between and. 5 5 (b) Their denominators are both 5. So into how many pieces do you need to divide the unit from 0 to? How many marks will you need to divide the unit into that many pieces? (c) Divide the unit from 0 to into five equal parts, and label the marks, consecutively,,,, (d) Now put points to mark and 5 5. ) The only mixed number to plot is. (a) Between which two whole numbers is ? Remember that a mixed number is a whole number plus a proper fraction, so. Since it is greater than three, but not a whole unit greater, is between and. (b) Divide that portion of the number line into equal pieces (thirds) by making marks. This file is copyright 07, Rice University. All Rights Reserved.

14 (c) Plot at the first mark ) Finally, look at the improper fractions,,. Locating these points will be easier if you change each of them to a mixed number. (a) (b) (c) ) Here is the number line with all the points (,,,, 7, 9, 5, and that your number line looks the same. ) plotted. Verify ) Locate and label the fractions 5 7,,,, 5 on the number line below Now let s locate some negative fractions. 7) We ll locate first. Remember that negative numbers are opposites of positive numbers, so is the opposite of. (a) Since is between the two whole numbers and, is between the two integers and. (b) Into how many pieces do we need to divide the unit between 0 and? (c) Divide that portion of the number line into equal pieces (halves) by making marks. This file is copyright 07, Rice University. All Rights Reserved.

15 (d) Plot at the mark ) Now let s locate on a number line. (a) Think about (b) So first. It is located between the whole numbers and. is between and. (c) Into how many equal pieces do we need to divide that unit? (d) Plot at the first mark. 9) Locating (a) on a number line will be easier if you first change it to a mixed number. =. It is between and. (b) Plot ) Locate and label the fractions and on the number line below ) Locate and label the fractions 9 and 9 on the number line below ) Locate and label the fractions 7,,, and 8 5 on the number line below ) Locate and label the fractions 5 7,,,, and 5 on the number line below This file is copyright 07, Rice University. All Rights Reserved.

16 Manipulative Mathematics Number Line Part Extra Practice Name Name the coordinate of each point. ) 0 ) 0 ) 0 ) 5) 5 Locate and label each point on the number line. 6) (a) 5 (b) (c) ) (a) 0 (b) 5 (c) This file is copyright 07, Rice University. All Rights Reserved.

17 8) (a) (b) (c) ) (a) 5 (b) (c) ) (a) 5 8 (b) (c) You can do more practice locating fractions on the number line at the website 6 This file is copyright 07, Rice University. All Rights Reserved.

18 Manipulative Mathematics Multiplication/Factors Name ) Take twelve tiles and form a rectangle. (a) How many rows does your rectangle have? (b) How many columns? (c) Each rectangle can be called a number of rows number of columns rectangle. The number of rows and the number of columns are called the dimensions of the rectangle. rows by columns rectangle (d) How many tiles were used to make this rectangle? (e) What is the product of 6? (f) Form a 6 rectangle. Draw it here: (g) What do you notice about the 6 and the 6 rectangles? A 6 rectangle and a 6 rectangle are equivalent. This means you could rotate the 6 rectangle and it would look exactly the same as the 6 rectangle. ) Now, create all possible rectangles using tile, tiles, tiles,, 5 tiles. (a) Copy each rectangle onto graph paper. (b) Label each rectangle with the total number of tiles used to form it. (c) Under the rectangle write its dimensions:. number of rows number of columns For example, your graph paper would show rectangles for tiles: 6 7 This file is copyright 07, Rice University. All Rights Reserved.

19 (d) Summarize your results in the chart below. Number of tiles Dimensions of the rectangles formed , 6, Use your chart to answer the following questions. ) Look for all the rectangles in your chart that have rows. (a) List the dimensions of all the rectangles that have rows.,,,,,,,,,,, (b) Now list the total number of tiles you used to form each rectangle you listed in 5.,,,,,,,,,,, These numbers are called the multiples of. Multiple A number is a multiple of n if it is the product of a counting number and n. 8 This file is copyright 07, Rice University. All Rights Reserved.

20 ) How can you use your rectangle chart to find the multiples of? 5) List the multiples of three:,,,,,,,. 6) List the multiples of four:,,,,,. 7) List the multiples of five: 5,,,,. Notice that with tiles, we could form different rectangles,, 6, and. The numbers,,,, 6, and are factors of, because, 6, and =. Factors If a b m, then a and bare factors of m. 8) List all the factors of 5:,,, 9) Which number from to 5 has the most factors? 0) Which number of tiles can be used to make the most rectangles? ) Explain why some numbers can be used to make more rectangles than other numbers. ) List the numbers for which you could only form one rectangle. These numbers are called primes. A prime number has only two factors, and itself. Prime A prime number is a counting number greater than, whose only factors are and itself. ) List all the primes between and 5. ) What other number relationships do you notice in your rectangle chart? 9 This file is copyright 07, Rice University. All Rights Reserved.

21 Manipulative Mathematics Multiplication/Factors Extra Practice Name List the first ten multiples of the following numbers. ) 6:,,,,,,,,, ) 7:,,,,,,,,, ) 8:,,,,,,,,, ) 9:,,,,,,,,, 5) :,,,,,,,,, The number 6 can be factored 6, 8, and, so all the factors of 6 are,,, 8, and 6. Find all the factors of each of the following numbers. 6) 7) 0 8) 9) 6 0) 5 0 This file is copyright 07, Rice University. All Rights Reserved.

22 Manipulative Mathematics Square Numbers Name ) Put about 50 color counters on your workspace. We will use the counters to make squares. (a) For example, is a square made from counters. It has counters on each side. (b) Make as many squares as you can with your counters. Draw a picture of each square that you create and record your results in the table below: Picture of square Total number of counters in the square Number of counters on each side ) Can you make a square with exactly 6 counters? Why or why not? ) Imagine if you had 00 counters (a) Could you make a square with exactly 00 counters? (b) Why or why not? (c) How many counters would be on each side of a square made with 00 counters? ) Work with a partner and put all your counters together. (a) Create a square that uses more than 50 counters. Draw a sketch of your square. This file is copyright 07, Rice University. All Rights Reserved.

23 (b) Create all the squares you can using 50 to 00 counters. Sketch your squares here. When a number n is multiplied by itself, we write it n and read it n squared. For example, 8 is read 8 squared. 6 is called the square of 8. Similarly, is the square of, because is. Square of a number If n m, then m is the square of n. 5) Complete this table to show the squares of the counting numbers through 5. n n 6 The squares of the counting numbers are called perfect squares, so the second row of the table shows the first fifteen perfect squares. 6) List the total number of counters you used for each square you made in Exercise (b). 7) Do you see a similarity between the table you filled in for Exercise with the pictures of squares and the table you made in Exercise 5 with the squares of the counting numbers through 5? (a) Describe how the two tables are alike. (b) Why do we use the word square for both the symbol in and the shape? This file is copyright 07, Rice University. All Rights Reserved.

24 Manipulative Mathematics Square Numbers Extra Practice Name Identify whether or not each number is a perfect square. If it is a perfect square, write is as the square of a counting number. Number Not a perfect square Yes perfect square ) 6 ) 50 ) 0 ) 96 5) 6) 89 7) 6 8) 65 9) 78 0) 96 This file is copyright 07, Rice University. All Rights Reserved.

25 Manipulative Mathematics Model Fractions Name Fraction: A fraction is written a b a is the numerator and b is the denominator. Fractions are a way to represent parts of a whole. The fraction means that one whole has been divided into equal parts and each part is one of the three equal parts. ) This circle that has been divided into equal parts. Label each part. ) What does the fraction represent? This means the whole has been divided into equal parts, and represents two of those three parts. Shade two out of the three parts of this circle to represent. ) What fraction of this circle is shaded? (a) How many parts are shaded? (b) How many equal parts are there? (c) The fraction of the circle that is shaded is. ) What fraction of this square is shaded? (a) How many parts are shaded? (b) How many equal parts are there? (c) The fraction of the square that is shaded is 5) To shade of the circle, shade out of the parts. Shade. This file is copyright 07, Rice University. All Rights Reserved.

26 Manipulative Mathematics Model Fractions Extra Practice Name Name the fraction modeled by each figure. ) ) ) ) 5) 6) 7) 8) Model each fraction. 9) 6 0) 5 9 ) 5 ) 7 8 For more practice naming fractions, go to: t_.html?from=topic_t_.html modeling fractions, go to: t_.html?from=topic_t_.html 5 This file is copyright 07, Rice University. All Rights Reserved.

27 Manipulative Mathematics Fractions Equivalent to One Name Fractions are often shown as parts of rectangles. Here, the whole is one long rectangle. Set up your fraction tiles as shown in the diagram above. ) How many of the tiles does it take to make whole tile? (a) It takes halves to make a whole. (b) Two out of two is whole.. ) How many of the tiles does it take to make whole tile? (a) It takes thirds to make a whole. (b) Three out of three is whole.. ) How many of the tiles does it take to make whole tile? (a) It takes fourths to make whole. (b) Four out of four is whole.. How many of the 6 tiles does it take to make whole tile? (a) It takes sixths. (b) Six out of six is whole What if the whole was divided into equal parts? We don t have fraction tiles to represent this and it is too many to draw easily, but try to visualize it in your mind. (a) How many s does it take to make? (b) ) Do you see any pattern here? Describe the pattern you see. 6 This file is copyright 07, Rice University. All Rights Reserved.

28 Manipulative Mathematics Name Fractions Equivalent to One Extra Practice Use fraction tiles to answer these exercises. You may want to use virtual fraction tiles on the interactive website ) How many 5 s does it take to make? ) How many 8 s does it take to make? ) How many 0 ) How many 5) How many 6) How many 6 s does it take to make? s does it take to make? s does it take to make? s does it take to make? 7) Fill in each numerator. (a) (b) (c) 9 8) Fill in each denominator. 8 (a) (b) (c) 5 9) Fill in the missing part. 0 (a) (b) (c) (d) (e) (f) 00 7 This file is copyright 07, Rice University. All Rights Reserved.

29 Manipulative Mathematics Mixed Numbers and Improper Fractions Name ) Use fraction circles to make wholes, if possible, with the following pieces. Draw a sketch to show your result. (a) halves (b) 6 sixths (c) fourths (d) 5 fifths ) Use fraction circles to make wholes, if possible, with the following pieces. Draw a sketch to show your result. (a) halves (b) 5 fourths (c) 8 fifths (d) 7 thirds When a fraction has the numerator smaller than the denominator, it is called a proper fraction. Its value is less than one. Fractions like,, and 7 8 A fraction like 5,, 8, or 7 5 denominator. Its value is greater than one. Proper and Improper Fractions are proper fractions. is called an improper fraction. Its numerator is greater than its The fraction a b is: ( b 0) proper if a b or improper if a b ) Write as improper fractions. (a) halves (b) 5 fourths (c) 8 fifths (d) 7 thirds 8 This file is copyright 07, Rice University. All Rights Reserved.

30 ) Look back at your models in Exercise and the improper fractions in Exercise. Which improper fraction in Exercise could also be written as? The number called a mixed number; it consists of a whole number and a proper fraction. Mixed Number A mixed number is written b a c c 0 A mixed number consists of a whole number a and a proper fraction b c. The model shows that 5 has the same value as. 5 5) Write each improper fraction as a mixed number. You may want to refer to your models in Exercise. (a) (b) 5 (c) 8 5 (d) 7 6) Rewrite the improper fraction 6 as a mixed number. Use fraction circles to find the result. (a) Draw a sketch to show your answer. (b) 6 7) Rewrite the improper fraction 7 5 as a mixed number. Use fraction circles to find the result. (a) Draw a sketch to show your answer. (b) This file is copyright 07, Rice University. All Rights Reserved.

31 8) Explain how you convert an improper fraction as a mixed number. 9) Rewrite the mixed number as an improper fraction. (a) Draw a sketch to show your answer. (b) 0) Rewrite the mixed number as an improper fraction. (a) Draw a sketch to show your answer. (b) ) Explain how you convert a mixed number to an improper fraction. 0 This file is copyright 07, Rice University. All Rights Reserved.

32 Manipulative Mathematics Name Mixed Numbers and Improper Fractions Extra Practice Use sets of fraction circles to do these exercises. You may want to use the fraction circles on the interactive website t_.html?open=activities&from=topic_t_.html. Name each improper fraction. Then write each improper fraction as a mixed number. ) ) (a) improper fraction (a) improper fraction (b) mixed number (b) mixed number Draw a figure to model the following improper fractions. Then write each as a mixed number. Improper fraction ) 7 Model Mixed number 7 ) ) ) 0 0 This file is copyright 07, Rice University. All Rights Reserved.

33 Draw a figure to model the following mixed numbers. Then write each as an improper fraction. Mixed number 7) 5 Model 5 Improper fraction 8) 6 6 9) 7 7 0) This file is copyright 07, Rice University. All Rights Reserved.

34 Manipulative Mathematics Equivalent Fractions Name Equivalent Fractions Equivalent fractions have the same value. Use fraction tiles to do the following activity: ) Take one of the tiles and set it on your workspace. (a) How many fourths equal one-half? Take the tiles and place them below the tile. How many of the tiles exactly cover the? (b) Since of the tiles cover the tile, we see is the same as. ) How many sixths equal one-half? (a) How many of the 6 tiles exactly cover the tile? (b) Draw a sketch to show your result. (c) Since of the 6 tiles cover the tile, we see is the same as 6. 6 ) How many eighths equal one-half? Draw a figure that demonstrates your answer. 8 This file is copyright 07, Rice University. All Rights Reserved.

35 ) How many tenths equal one-half? Draw a figure that demonstrates your answer. 0 5) How many twelfths equal one-half? Draw a figure that demonstrates your answer 0 6) Suppose you had bars marked. How many of them would it take to equal one-half? 0 Take one of the bars and set it on your workspace. 7) How many sixths equal one-third? Draw a figure that demonstrates your answer. 6 8) How many twelfths equal one-third? Draw a figure that demonstrates your answer. 9) Suppose you had tiles marked. 0 How many of them would it take to equal one-third? 0 0) How many sixths equal two-thirds? Draw a figure that demonstrates your answer. 6 This file is copyright 07, Rice University. All Rights Reserved.

36 ) How many eighths equal three-fourths? Draw a figure that demonstrates your answer. 8 ) How many twelfths equal three-fourths? Draw a figure that demonstrates your answer. ) Suppose you had tiles marked. 0 (a) How many of them would it take to equal seven-tenths? (b) Explain how you got your answer. ) Can you use twelfths to make a fraction equivalent to three-fifths? Explain your reasoning. 5 This file is copyright 07, Rice University. All Rights Reserved.

37 Manipulative Mathematics Equivalent Fractions Extra Practice Name Use fraction tiles to do these exercises. You may want to use virtual fraction tiles on the interactive website ) How many eighths equal one-fourth? Draw a figure that demonstrates your answer. 8 ) How many twelfths equal one-third? Draw a figure that demonstrates your answer. ) How many tenths equal four-fifths? Draw a figure that demonstrates your answer. 0 5 ) How many sixteenths equal three-fourths? Draw a figure that demonstrates your answer. 6 5) How many fifteenths equal two-thirds? Draw a figure that demonstrates your answer. 5 6) How many fifteenths equal two-fifths? Draw a figure that demonstrates your answer ) How many twelfths equal six-eighths? Draw a figure that demonstrates your answer ) How many twelfths equal six-ninths? Draw a figure that demonstrates your answer. 6 This file is copyright 07, Rice University. All Rights Reserved. 6 9

38 Manipulative Mathematics Model Fraction Multiplication Name When you multiply fractions, do you need a common denominator? Do you take the reciprocal of one of the fractions? What are you supposed to do and how are you going to remember it? A model may help you understand multiplication of fractions. ) Model the product. (a) To multiply and, let s think of. (b) First, we draw a rectangle to represent one whole. We divide it vertically into equal parts, and then shade in three of the parts to model. We have shaded in of the rectangle. (c) Now, we divide the rectangle horizontally into two equal parts to divide the whole into halves. Then we double-shade of what was already shaded. (d) Into how many equal pieces is the rectangle divided now? (e) How many of these pieces are double-shaded? We double-shaded out of the 8 equal pieces, of the rectangle. So of is. 8 8 We showed that 8 Notice multiplying the numerators multiplying the denominators 8 7 This file is copyright 07, Rice University. All Rights Reserved.

39 ) Model the product. 5 (a) To multiply and, think of. 5 (b) First shade in 5 of the rectangle. (c) Now double-shade of what was already shaded. (d) Into how many equal pieces is the rectangle divided now? (e) How many pieces are double-shaded? (f) What fraction of the rectangle is double-shaded? (g) So of is. 5 You have shown that 5 0 Notice multiplying the numerators multiplying the denominators 5 0 ) Use a rectangle to model each product. Sketch a diagram to illustrate your model. (a) 8 This file is copyright 07, Rice University. All Rights Reserved.

40 (b) (c) (d) (e) 5 5 ) Look at each of your models and answers in Question. (a) If you multiply numerators and multiply denominators, do you get the same result as you did from the model? (b) Explain in words how to multiply two fractions. 5) The definition of fraction multiplication is given in the box below. Fraction Multiplication a c ac If a, b, c, and d are numbers where b 0and d 0, then. b d bd To multiply fractions, multiply the numerators and multiply the denominators. 5 7 Use the definition of fraction multiplication to multiply (a) Identify a, b, c, and d. (b) Multiply the fractions. 9 This file is copyright 07, Rice University. All Rights Reserved.

41 Manipulative Mathematics Model Fraction Multiplication Extra Practice Name Use a rectangle to model each multiplication. Sketch your model and write the product. You can practice using rectangles to model fraction multiplication online at the website t_.html?from=search.html?qt=multiply+fractio ns. ) ) 6 8 ) ) 5 5) 6) Multiply. 7) 8) ) 0) This file is copyright 07, Rice University. All Rights Reserved.

42 Manipulative Mathematics Model Fraction Division Name Model Fraction Division ) Why is? Let s model this with counters. (a) How many groups of counters can be made from the shown below? (b) Draw a circle around each group of counters. How many groups of counters do you have? (c) There are groups of counters. In other words, there are s in. So,. What about dividing fractions? Get out your fraction tiles and let s see! ) To model the quotient with fraction tiles we want to see how many sixths there are in 6 one-half. (a) Line up your half and sixth fraction tiles as shown below (b) How many s are in? 6 (c) 6 ) Model the quotient with fraction tiles. 8 Use your fourth and eighth fraction tiles to find out how many eighths there are in one fourth. (a) Draw a sketch of your result here. This file is copyright 07, Rice University. All Rights Reserved.

43 (b) There are s in 8. (c) So 8 ) Model the quotient with fraction tiles 6 Use your third and sixth fraction tiles to find out how many sixths there are in one third a) Draw a sketch of your result here. b) There are s in. c) So 6 6 5) Model the quotient with fraction tiles 8 Use your half and eighth fraction tiles to find out how many eighths there are in one half. a) Draw a sketch of your result here. b) There are s in c) So 8 8 Model a Whole Number Divided by a Fraction 6) Use fraction bars to model the quotient (a) How many s are there in? This file is copyright 07, Rice University. All Rights Reserved.

44 (b) There are s in, so (c) Let s think of this example another way in terms of money. We often read as one quarter, so you can think of, as asking how many quarters are there in two dollars? We know that $ is quarters, so how many quarters are in $? (d) So,. 7) Use fraction tiles to model the following. Sketch a diagram to illustrate your model. a) b) 6 Using fraction tiles in exercise, we showed that. Notice that also. 6 6 How does relate to? They are reciprocals! To divide fractions, we multiply the first 6 fraction by the reciprocal of the second. This leads to the following definition. Fraction Division If a, b, c, and d are numbers where b 0, c 0 and d 0, then a c a d b d b c 5 8) Use the Fraction Division definition above to find the quotient. 7 8 (a) Identify the numbers that correspond to a, b, c, and d. This file is copyright 07, Rice University. All Rights Reserved.

45 (b) Divide the fractions. 9) Explain in words how to divide two fractions. 0) Explain in words how to divide a whole number by a fraction. This file is copyright 07, Rice University. All Rights Reserved.

46 Manipulative Mathematics Name Model Fraction Division Extra Practice Use fraction tiles to model each division. Sketch your model and write the quotient. You may want to use the fraction tiles shown at the interactive website: ) ) 0 ) ) 5) 6) Divide. 5 7) 8) ) 0) This file is copyright 07, Rice University. All Rights Reserved.

47 Manipulative Mathematics Model Fraction Addition Name How many quarters are pictured above? One quarter plus quarters equals quarters. Quarters? Remember, quarters are really fractions of a dollar; quarter is another word for fourth. So the picture of the coins shows that. Let s use fraction circles to model addition of fractions for the same example,. Start with one piece. Add two more pieces. + The result is. So,. ) Use fraction circles to model the sum. 8 8 (a) Take three pieces. Add two more pieces. How many pieces do you have? (b) Sketch your model here. (c) You have five eighths. 8 8 ) Use fraction circles to model the following. Sketch a diagram to illustrate your model. (a). (b) 6 6 (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. 6 This file is copyright 07, Rice University. All Rights Reserved.

48 ) Use fraction circles to model the following. Sketch a diagram to illustrate your model. (a) (b) (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. ) Use fraction circles to model the following. Sketch a diagram to illustrate your model. (a) (b) (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. 5) A common error made by students when adding fractions is to add the numerators and add the denominators (much like we multiply numerators and multiply denominators when multiplying fractions). Use a model to see why this does not work for addition! (a) Model. Sketch a diagram to illustrate your model. 5 5 (b) Did the fifths change to another size piece? Did they change to (c) pieces? 7 This file is copyright 07, Rice University. All Rights Reserved.

49 These examples show that to add the same size fraction pieces that is, fractions with the same denominator you just add the number of pieces. So, to add fractions with the same denominator, you add the numerators and place the sum over the common denominator. This leads to the following definition. Fraction Addition If a, b, and c are numbers where c 0, then a b a b c c c 6 8 6) Use the definition of fraction addition in the box above to add. (a) Identify a, b, and c. (b) Add the fractions.. 7) Explain in words how to add two fractions that have the same denominator. 8 This file is copyright 07, Rice University. All Rights Reserved.

50 Manipulative Mathematics Model Fraction Addition Extra Practice Name Use fraction circles to model each addition. Sketch your model and write the sum. You may want to use the fraction circles on the interactive website: t_.html?open=activities&hidepanel=true&from =topic_t_.html. ) ) ) ) 5) 6) ) 8) 9) ) ) ) This file is copyright 07, Rice University. All Rights Reserved.

51 Manipulative Mathematics Model Fraction Subtraction Name Subtracting two fractions with common denominators works the same as addition of fractions with common denominators. Think of a pizza that was cut into twelve equal slices. Each piece is 7 of the pizza. After dinner there are seven pieces,, left in the box. If Leonardo eats of the pieces,, how much is left? There would be 5 pieces left,. So. 7 ) Let s use Fraction Circles to model the same example,. (a) Start with seven pieces. Take away two pieces. How many twelfths do you have left? 5 (b) You have five pieces left,. ) Use your fraction circles to model the difference. 5 5 Start with four pieces. Take away one piece. 5 5 (a) How many fifths do you have left? (b) Sketch your model here. 7 (c) You have fifths left. 5 5 ) Use fraction circles to model the following. Sketch a diagram to illustrate your model. 7 5 (a) (b) (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. 50 This file is copyright 07, Rice University. All Rights Reserved.

52 ) Use fraction circles to model the following. Sketch a diagram to illustrate your model. (a) (b) 5 5 (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. 5) Use fraction circles to model the following. Sketch a diagram to illustrate your model. 5 7 (a) (b) (c) Look at parts (a) and (b). Explain how you got the numerator and denominator of your answers. These examples show that to subtract the same size fraction pieces that is, fractions with the same denominator you just subtract the number of pieces. So, to subtract fractions with the same denominator, you subtract the numerators and place the difference over the common denominator. This leads to the following definition. Fraction Subtraction If a, b, and c are numbers where c 0, then a b a b c c c 5 6) Use the definition of fraction subtraction in the box above to subtract. 7 7 (a) Identify a, b, and c. (b) Subtract the fractions. 7) Explain in words how to subtract two fractions that have the same denominator. 5 This file is copyright 07, Rice University. All Rights Reserved.

53 Manipulative Mathematics Model Fraction Subtraction Extra Practice Name Use fraction circles to model each subtraction. Sketch your model and write the difference. You may want to use the fraction circles on the interactive website: t_.html?open=activities&hidepanel=true&from =topic_t_.html. 5 ) ) ) ) 5) 6) ) 8) 9) ) ) ) This file is copyright 07, Rice University. All Rights Reserved.

54 Manipulative Mathematics Name Model Finding the Least Common Denominator Let s look at coins again. Can you add one quarter and one dime? Well, you could say there are two coins, but that s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit cents. One quarter equals 5 cents and one dime equals 0 cents, so the sum is 5 cents. One quarter + one dime Similarly, when you add fractions with different denominators you have to convert them to equivalent fractions with a common denominator. With the coins, when we converted to cents, the denominator was cents is and 0 cents is and so we added to get , which is 5 cents. 00 Use fraction pieces to find the least common denominator of and. Take out your set of fraction pieces and place and on your workspace. You need to find a common fraction piece that can be used to cover both and exactly. ) Try the pieces. fraction circles fraction tiles (a) Can you cover the piece exactly with pieces? (b) How many pieces cover the piece? (c) Can you cover the piece exactly with pieces? (d) How many pieces cover the piece? (e) Sketch your results here. 5 This file is copyright 07, Rice University. All Rights Reserved.

55 ) Try the pieces. 5 (a) Can you cover the piece exactly with pieces? 5 (b) How many pieces cover the piece? 5 (c) Can you cover the piece exactly with pieces?. 5 (d) How many pieces cover the piece? 5 (e) Sketch your results here. ) Try the pieces. 6 (a) Can you cover the piece exactly with pieces? 6 (b) How many pieces cover the piece? 6 (c) Can you cover the piece exactly with pieces?. 6 (d) How many pieces cover the piece? 6 (e) Sketch your results here. ) You have shown that: (a) of the pieces exactly cover the 6 piece. (b) of the pieces exactly cover the 6 piece. 6 6 The smallest denominator of a fraction piece that can be used to cover both fractions exactly is the least common denominator (LCD) of the two fractions. The smallest denominator of a fraction piece that can be used to cover both and is 6. So, you have found that the least common denominator of and is 6. 5 This file is copyright 07, Rice University. All Rights Reserved.

56 Use fraction pieces to find the least common denominator of and. Place and on 6 6 your workspace. Find a common fraction piece that can be used to cover both and 6 exactly. 5) Sketch your results here. 6) You have shown that: (a) of the pieces exactly cover the piece. (b) of the pieces exactly cover the 6 piece. 6 (c) Both fractions can be written with denominator, so is their common denominator. Use fraction pieces to find the least common denominator of and. Find a common fraction piece that can be used to cover both and exactly. 7) Sketch your results here. 8) You have shown that: (a) of the pieces exactly cover the piece. (b) of the pieces exactly cover the piece. (c) Both fractions can be written with denominator, so is their common denominator. 55 This file is copyright 07, Rice University. All Rights Reserved.

57 Use fraction pieces to find the least common denominator of and. Find a common 5 fraction piece that can be used to cover both and exactly. 5 9) Sketch your results here. 0) You have shown that: (a) of the pieces exactly cover the piece. (b) of the pieces exactly cover the 5 piece. 5 (c) Both fractions can be written with denominator, so is their common denominator. 56 This file is copyright 07, Rice University. All Rights Reserved.

58 Manipulative Mathematics Name Model Finding the Least Common Denominator Extra Practice Use fraction tiles or fraction circles to find the least common denominator (LCD) of each pair of fractions, and to re-write each fraction with the LCD. Sketch your model. You may want to use the fraction tiles on the interactive website: or the fraction circles at t_.html?open=activities&hidepanel=true&from =topic_t_.html to work these exercises. ) and (a) LCD = ) and 6 8 (a) LCD = (b) = (b) = (c) = (c) 6 8 = (d) sketch your model. (d) sketch your model. ) and (a) LCD = ) and (a) LCD = (b) = (b) = (c) = (c) = (d) sketch your model. (d) sketch your model. 5 5) and (a) LCD = 6) and 8 6 (a) LCD = 5 (b) = (b) 8 6 = (c) = (c) = (d) sketch your model. (d) sketch your model. 57 This file is copyright 07, Rice University. All Rights Reserved.

59 7) and (a) LCD = 8) and 5 (a) LCD = (b) = (b) = (c) = (c) 5 = (d) sketch your model. (d) sketch your model ) and (a) LCD = 0) and 6 (a) LCD = 5 (b) = (b) = 5 (c) = (c) 6 = (d) sketch your model. (d) sketch your model. 58 This file is copyright 07, Rice University. All Rights Reserved.

60 Manipulative Mathematics Addition of Signed Numbers Name We are going to model signed numbers with two-color counters. One white counter,, will represent one positive unit. One red counter,, will represent one negative unit. When we have one positive and one negative together, of a neutral pair is zero. we call it a neutral pair. The value ) We ll start by modeling 5, the sum of 5 and. (a) Start with 5 positives. (b) Add positives. Put counters of the same color in the same row. (c) How many counters are there? positives 5 8 ) Now we ll model 5 ( ), the sum of negative 5 and negative. (a) Start with 5 negatives. (b) Add negatives. (c) How many counters are there? negatives 5 8 ) What about adding numbers with different signs? Let s model 5, the sum of negative 5 and. (a) Start with 5 negatives. (b) Add positives. Since they are a different color, line them up under the red counters. (c) Are there any neutral pairs? Remove the neutral pairs. (d) How many are left? negatives 5 ) The fourth case is the sum of a positive and a negative. We ll model 5 ( ), the sum of 5 and negative. (a) Start with 5 positives. (b) Add negatives. 59 This file is copyright 07, Rice University. All Rights Reserved.

61 (c) Remove the neutral pairs. (d) How many are left? 5 Use your counters to model each sum. Draw a sketch of your model. 5) 6) 5 ( 5) 7) 8) ( ) 9) 8 ( ) 0) 7 ( ) ) ( ) ) 5 7 ) ( ) ) 5) 7 ( ) 6) 7) Do you notice a pattern? Explain in words how to add: (a) 8 ( 0) (b) 5 5 8) Without using counters, try to find these sums. (a) 5 9 (b) 57 ( ) (c) 78 ( 7) (d) 6 60 This file is copyright 07, Rice University. All Rights Reserved.

62 Manipulative Mathematics Name Addition of Signed Numbers Extra Practice Use two-color counters to model each addition. You can find virtual counters on the website: t_.html?from=topic_t_.html. If you use the website, click on User at the bottom of the workspace so that you can enter the numbers in each exercise. Sketch the model for each addition and find the sum. ) 5 ) 5 ( ) ) 5 ) 5 ( ) 5) 6 ( 6) 6) ( ) 7) ( 5) 8) 6 8 9) ( 7) 0) ) ( 8) ) 9 6 This file is copyright 07, Rice University. All Rights Reserved.

63 Manipulative Mathematics Subtraction of Signed Numbers Name We are going to model signed numbers with two-color counters. One white counter,, will represent one positive unit. One red counter,, will represent one negative unit. When we have one positive and one negative together, of a neutral pair is zero. we call it a neutral pair. The value ) We ll start by modeling 5, the difference of 5 and. (a) Start with 5 positives. (b) Take away positives. (c) How many counters are left? positives 5 ) Now we ll model 5 ( ), the difference of negative 5 and negative. (a) Start with 5 negatives. (b) Take away negatives. (c) How many counters are left? negatives 5 ) What about subtracting numbers with different signs? Let s model 5, the difference of negative 5 and. (a) Start with 5 negatives. (b) We want to take away positives. Do we have any positives to take away? (c) We can add neutral pairs to get the positives. (d) Now take away positives. (e) How many counters are left? ) The fourth case is the sum of a positive and a negative. We ll model 5 ( ), the difference of 5 and negative. (a) Start with 5 positives. 6 This file is copyright 07, Rice University. All Rights Reserved.

64 (b) We want to take away negatives. Do we have any negatives to take away? (c) But we can add neutral pairs to get the negatives. (d) Take away negatives. (e) How many counters are left? Use your counters to model each difference. Draw a sketch of your model. 5) 7 6) 6 ( ) 7) 8) ( ) 9) ( ) 0) 5 ( ) ) 8 6 ) 7 ) ( ) ) 5) 5 6) 6 7) Do you notice a pattern? Explain in words how to subtract: (a) 8 ( ) (b) 0 5 8) Without using counters, try to find these differences. (a) 5 9 (b) 57 ( ) (c) 78 ( 7) (d) 6 6 This file is copyright 07, Rice University. All Rights Reserved.

65 Manipulative Mathematics Name Subtraction of Signed Numbers Extra Practice Use two-color counters to model each subtraction. You can find virtual counters on the website t_.html?from=topic_t_.html. If you use the website, click on User at the bottom of the workspace so that you can enter the numbers in each exercise. Sketch the model for each subtraction and find the difference. ) 7 ) 7 ( ) ) 7 ) 7 ( ) 5) 6 ( 5) 6) ( ) 7) 8 8 8) 9 5 9) ( ) 0) 5 ) 6 ) 0 6 This file is copyright 07, Rice University. All Rights Reserved.

66 Manipulative Mathematics Multiples Name Multiple of a Number A number is a multiple of n if it is the product of a counting number and n. ) Multiples of (a) This table lists the counting numbers from to 50. Highlight all the multiples of (b) Now look at all the numbers that you highlighted. Describe a pattern you notice. (c) Create a rule you could use to determine if a number larger than 50 is a multiple of. (d) Use your rule to decide if 97 is a multiple of. (e) Is 86 a multiple of? ) Multiples of 5 (a) This table lists the counting numbers from to 50. Highlight all the multiples of (b) Now look at all the numbers that you highlighted. Describe a pattern you notice. (c) Create a rule you could use to determine if a number larger than 50 is a multiple of 5. (d) Use your rule to decide if 7 is a multiple of This file is copyright 07, Rice University. All Rights Reserved.

67 (e) Is 90 a multiple of 5? ) Multiples of 0 (a) The table lists the counting numbers from to 50. Highlight all the multiples of (b) Now look at all the numbers that you highlighted. Describe a pattern you notice. (c) Create a rule you could use to determine if a number larger than 50 is a multiple of 0. (d) Use your rule to decide if 690 is a multiple of 0. (e) Is 875 a multiple of 0? ) Multiples of (a) The table lists the counting numbers from to 50. Highlight all the multiples of (b) List the multiples of. (c) Under each multiple of, find the sum of the digits of that number. For example, is a multiple of, and 6. What do you notice about all the multiples of? sumof digits 6, 6, 9,, 5, 8,...,, (d) Use these results to create a rule to determine if a number is a multiple of. (e) Use your rule to decide if 75 is a multiple of. (f) Is 88 a multiple of? 66 This file is copyright 07, Rice University. All Rights Reserved.

68 Manipulative Mathematics Multiples Extra Practice Name ) State a rule you can use to determine if a number is a multiple of: (a) (b) (c) 5 (d) 0 For each number, determine if it is a multiple of,, 5, and/or 0, and indicate your answers by writing yes or no in the spaces below. (a) multiple of (b) multiple of (c) multiple of 5 (d) multiple of 0 ) 0 (a) (b) (c) (d) ) 65 (a) (b) (c) (d) ) 5 (a) (b) (c) (d) 5) (a) (b) (c) (d) 6) 55 (a) (b) (c) (d) 7) 70 (a) (b) (c) (d) 8) 65 (a) (b) (c) (d) 9) 55 (a) (b) (c) (d) 0) 650 (a) (b) (c) (d) 67 This file is copyright 07, Rice University. All Rights Reserved.

69 Manipulative Mathematics Prime Numbers Name Prime Number A prime number is a counting number greater than, whose only factors are one and itself. A counting number that is not prime is composite. ) Use this table to find the primes less than 50. Remember a prime number is a number whose only factors are and itself. The number is not considered prime, so the smallest prime number is. (a) On the table, circle and then cross out all the multiples of. All multiples of, greater than, have two as a factor and so are not prime. (b) Next, circle and then cross out all the multiples of. All multiples of, greater that, have three as a factor and so are not prime. (c) Go to the next number that has not been crossed out. Circle it it is prime and then cross out all its multiples. (d) Continue this routine until all the numbers in the table have been crossed out or circled ) The numbers that have been crossed out are not prime. Counting numbers that are not prime are called. ) The circled numbers are prime. List the primes less than 50. ) What are the only factors of each prime you listed? 5) State one fact you notice about the primes. 68 This file is copyright 07, Rice University. All Rights Reserved.

70 Manipulative Mathematics Prime Numbers Extra Practice Name ) Use this table to find the primes less than 00. Remember a prime number is a number whose only factors are and itself. The number is not considered prime, so the smallest prime number is. (a) On the table, circle and then cross out all the multiples of. All multiples of, greater than, have two as a factor and so are not prime. (b) Next, circle and then cross out all the multiples of. All multiples of, greater than, have three as a factor and so are not prime. (c) Go to the next number that has not been crossed out. Circle it -it is prime and then cross out all its multiples. (d) Continue this routine until all the numbers in the table have been crossed out or circled ) The circled numbers are prime. List the primes less than 00. For more practice online, you can use the hundreds chart at t_.html?open=instructions&hidepanel=true&fr om=topic_t_.html. Display 0 rows to show the numbers to 00 and click Remove Multiples at the bottom of the workspace to remove (instead of crossing out) the multiples of each prime. 69 This file is copyright 07, Rice University. All Rights Reserved.

71 Manipulative Mathematics Subtraction Property of Equality Name ) You are going to solve a puzzle. Use your envelopes and counters to recreate the picture below on your workspace. Both sides have the same number of counters, but some counters are hidden in the envelope. The goal is to discover how many counters are in the envelope. (a) How many counters are in the envelope? counters are in the envelope. (b) What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope? List the steps here. Perhaps you are thinking- the counters at the bottom left can be matched with on the right. Then I can take them away from both sides. That leaves five on the right-so there must be 5 counters in the envelope. Try this with your envelope and counters. (c) Each side of the workspace models an expression and the line in the middle represents the equal sign, so we can write an algebraic equation from this model. What algebraic equation is modeled by this picture? 70 This file is copyright 07, Rice University. All Rights Reserved. =

72 Let s write algebraically the steps we took to discover how many counters were in the envelope: x 8 We took away three from each side. And then we had left. x x 8 5 (d) Check: 8 Five in the envelope plus three more equals eight! ) Let s try this again! How many counters are in the envelope? Use your envelope and counters to recreate this picture. Now, move the counters to find out how many counters are in the envelope. (a) List the steps you took to find out how many counters were in the envelope. (b) What algebraic equation is modeled by this picture? x (c) We need to take away from each side. x 6 (d) There are counters in the envelope! x (e) Check: 6 Four in the envelope plus two more does equal six! ) How many counters are in this envelope? Use your envelope and counters to recreate this picture. Move the counters to discover how many counters are in the envelope. (a) Write the algebraic equation that is modeled by this picture. x (b) Take away from each side. x 5 (c) There are counters in the envelope! x (d) Check: 5 7 This file is copyright 07, Rice University. All Rights Reserved.

73 ) How many counters are in this envelope? Use your envelope and counters to recreate this picture. Move the counters to find the number of counters in the envelope. (a) Write the equation modeled by the envelope and counters. = (b) Show the steps you take, in words and algebra, to find the number of counters in the envelope. Words Algebra 5) How many counters are in this envelope? Use your envelopes and counters to recreate this picture. Move the counters as needed to find the number of counters in the envelope. (a) Write the equation modeled by the envelope and counters. = (b) Show the steps you take, in words and algebra, to find the number of counters in the envelope. Words Algebra 7 This file is copyright 07, Rice University. All Rights Reserved.

74 6) Model a similar equation for your partner. Have your partner figure out how many counters are in the envelope. (a) Sketch a picture of your model. (b) Show the algebra steps your partner took to find the number of counters in the envelope. 7) Have your partner model a similar equation for you. Figure out how many counters are in the envelope. (a) Sketch a picture of the model. (b) Show the algebra steps you took to find the number of counters in the envelope. With these puzzles we have modeled a method for solving one kind of equation. To solve each equation, we used the Subtraction Property of Equality. The Subtraction Property of Equality: For any real numbers a, b, and c, if a b, then a c b c. When you subtract the same quantity from both sides of an equation, you still have equality! 7 This file is copyright 07, Rice University. All Rights Reserved.

75 Manipulative Mathematics Name Subtraction Property of Equality Extra Practice #-6: For each figure: (a) Write the equation modeled by the envelopes and counters. (b) Show the steps you take, in words and algebra, to find the number of counters in the envelope. ) (b) Solution Words Algebra (a) Equation = ) (b) Solution Words Algebra (a) Equation = ) (b) Solution Words Algebra (a) Equation = 7 This file is copyright 07, Rice University. All Rights Reserved.

76 ) (b) Solution Words Algebra (a) Equation = 5) (b) Solution Words Algebra (a) Equation = 6) (b) Solution Words Algebra (a) Equation = 75 This file is copyright 07, Rice University. All Rights Reserved.

77 #7-8: Solve each equation using the Subtraction Property of Equality. 7) 8) x 5 x 5 x x x 0 0 x 9) 0) x 9 7 x x 9 7 x x x ) ) x 6 5 x 75 0 x 6 5 x 75 0 x x ) x 8 ) y ) u 7 6) v ) m ) n This file is copyright 07, Rice University. All Rights Reserved.

78 Manipulative Mathematics Division Property of Equality Name ) You are going to solve a puzzle. Use your envelopes and counters to recreate the picture below on your workspace. Both sides have the same total number of counters, but some counters are hidden in the envelopes. Both envelopes contain the same number of counters. The goal is to discover how many counters are in each envelope. (a) How many counters are in each envelope? counters are in each envelope. (b) What are you thinking? What steps are you taking in your mind to figure out how many counters are in each envelope? List the steps here. Perhaps you are thinking that you have to separate the counters on the right side into groups, because there are envelopes. So 6 counters divided into groups means there must be counters in each envelope. Try this with your envelopes and counters. (c) Each side of the workspace models an expression and the line in the middle represents the equal sign, so we can write an algebraic equation from this model. What algebraic equation is modeled by this picture? = (d) Let s write algebraically the steps we took to discover how many counters were in the envelope: x 6 77 This file is copyright 07, Rice University. All Rights Reserved.

79 We divided both sides of the equation by, x 6 So we have in each envelope. x (e) Check: 6 Three counters in each of two envelopes equals six! ) Here s another puzzle. How many counters are in each envelope? Use your envelopes and counters to recreate this picture. Now, move the counters to find out how many counters are in each envelope. (a) List the steps you took to find out how many counters are in each envelope. (b) What algebraic equation is modeled by this picture? (c) We need to divide the counters into groups. x (d) Divide each side by. x (e) There are counters in each envelope! x (f) Check: ) How many counters are in each envelope? Use your envelopes and counters to recreate this picture. Move the counters to discover how many counters are in each envelope. (a) Write the algebraic equation that would match this situation. x (b) Divide each side by. x 8 (c) There are counters in each envelope! x (d) Check: 8 78 This file is copyright 07, Rice University. All Rights Reserved.

80 ) How many counters are in each envelope? Use your envelopes and counters to recreate this picture. Move the counters to find the number of counters in the envelope. (a) Write the equation modeled by the envelopes and counters. x (b) Show the steps you take, in words and algebra, to find the number of counters in the envelope. Words Algebra 5) How many counters are in each envelope? Use your envelopes and counters to recreate this picture. Move the counters as needed to find the number of counters in the envelope. (a) Write the equation modeled by the envelopes and counters. x (b) Show the steps you take, in words and algebra, to find the number of counters in the envelope. Words Algebra 79 This file is copyright 07, Rice University. All Rights Reserved.

81 6) Model a similar equation for your partner. Have your partner figure out how many counters are in each envelope. (a) Sketch a picture of your model. (b) Show the algebra steps your partner took to find the number of counters in each envelope. 7) Have your partner model a similar equation for you. Figure out how many counters are in each envelope. (a) Sketch a picture of the model. (b) Show the algebra steps you took to find the number of counters in each envelope. With these puzzles we have modeled a method for solving one kind of equation. To solve each equation, we used the Division Property of Equality. : The Division Property of Equality For any real numbers a, b, c, and c 0, a b if a b, then. c c When you divide both sides of an equation by any non-zero number, you still have equality! 80 This file is copyright 07, Rice University. All Rights Reserved.

82 Manipulative Mathematics Name Division Property of Equality Extra Practice #-6: For each figure: (a) write the equation modeled by the envelopes and counters. (b) show the steps you take, in words and algebra, to find the number of counters in each envelope. ) (b) Solution Words Algebra (a) Equation x ) (b) Solution Words Algebra (a) Equation x ) (b) Solution Words Algebra (a) Equation x 8 This file is copyright 07, Rice University. All Rights Reserved.

83 ) (b) Solution Words Algebra (a) Equation x 5) (b) Solution Words Algebra (a) Equation x 6) (b) Solution Words Algebra (a) Equation x 8 This file is copyright 07, Rice University. All Rights Reserved.

84 #7-8: Solve each equation using the Division Property of Equality. 7) 8) x 6 x 6 x 6 x 6 x x 9) 0) 8x 6 5x 5 8x 6 5x 5 x x ) ) 9x 5 x 08 9x 5 x 08 x x ) 7x ) n 65 5) 9y 8 6) 5q 75 7) 80p 800 8) 0m This file is copyright 07, Rice University. All Rights Reserved.

85 Manipulative Mathematics Visualizing Area and Perimeter Name A color tile is a square that is inch on a side. If an ant walked around the edge of the tile, it would have walked inches. This distance around the tile is called the perimeter of the tile. The area of the tile is measured by determining how many square inches (or other unit) cover the tile. Since a color tile is a square that is inch on each side, its area is one square inch. inch inch inch inch Perimeter is inches. Area is square inch. ) Use tiles to make a shape like the one shown below. Notice that each tile must touch the other along one complete side. (a) What is the perimeter of this shape? Perimeter = (b) What is the area? Area = (c) Can you make any other shape using two tiles? (d) Can you find any other perimeter using two tiles? (e) Record your results in the chart in #5. ) Make all possible shapes with tiles. Keep in mind that rotations and flips are really the same shape! Sketch your shapes on your grid paper, and color or shade in the squares. (a) How many shapes did you make? (b) For each shape, find its perimeter. Write the perimeter next to each shape. (c) What is the area of each shape that you made? Write the area inside each shape. (d) Record your results in the chart in #5. 8 This file is copyright 07, Rice University. All Rights Reserved.

86 ) Now use tiles. Sketch all the possible shapes on your grid paper. (a) How many shapes did you make? (b) For each shape, find its perimeter. Write the perimeter next to each shape. (c) What is the area of each shape that you made? Write the area inside each shape. (d) Record your results in the chart in #5. ) Take 5 tiles. Sketch all the possible shapes on your grid paper. (a) How many shapes did you make? (b) For each shape, find its perimeter. Write the perimeter next to each shape. (c) List all the perimeters of the shapes with 5 tiles. (d) Was more than one shape possible for any perimeter? (e) What is the smallest perimeter possible using 5 tiles? Why? (f) What is the largest perimeter possible using 5 tiles? Why? (g) What is the area of each shape that you made? Write the area inside each shape. (h) List all the areas of the shapes with 5 tiles. (i) Record your results in the chart in #5. 5) Fill in the chart below to show your results from #-. Number of tiles Perimeters Found Areas Found inches square inch 5 6) Name one fact you learned about perimeter from this activity. 7) Name one fact you learned about area from this activity. 85 This file is copyright 07, Rice University. All Rights Reserved.

87 Manipulative Mathematics Name Visualizing Area and Perimeter Extra Practice Find the area and perimeter of each shape. ) ) ) area= area= area= perimeter= perimeter= perimeter= ) 5) 6) area= area= area= perimeter= perimeter= perimeter= 7) 8) 9) area= area= area= perimeter= perimeter= perimeter= 0) ) ) area= area= area= perimeter= perimeter= perimeter= For more practice, use color tiles to make your own shapes and then find the area and perimeter. You may want to use the square blocks at the interactive website: t_.html?open=activities&from=topic_t_.html 86 This file is copyright 07, Rice University. All Rights Reserved.

88 Manipulative Mathematics Measuring Area and Perimeter Name The area of a shape is measured by determining how many square inches (or other unit) cover the shape. The perimeter is the distance around the shape. A color tile is a square that is inch long on each side. Its area is one square inch. Its perimeter is inches. inch inch inch inch Perimeter is inches. Area is square inch. If we put two tiles side by side we have a shape with area two square inches. The perimeter is 6 inches, because the distance along a side of each square is inch. inches inch inch inches Perimeter is 6 inches. Area is square inches. ) Take your set of tiles and Shape l. (a) First, estimate how many tiles will be needed to completely cover Shape l. Record this in the Estimated Area column on the chart below. (b) Next, estimate how many tiles will form the perimeter of Shape I. Record this in the Estimated Perimeter column on the chart below. (c) Now cover Shape l completely with tiles. Count the number of tiles you used and record this in the Measured Area column in the chart on the next page. Count the number of tiles along the perimeter and record this in the Measured Perimeter column. 87 This file is copyright 07, Rice University. All Rights Reserved.

89 ) Repeat this process with the rest of your shapes. Shape I Estimated Area Estimated Perimeter Measured Area Measured Perimeter II III IV V VI ) Think about area. (a) When might you need to use area in your everyday life? (b) Give an example of when estimating an area is useful. (c) Give an example of when measuring an area is necessary. ) Think about perimeter. (a) When might you need to use perimeter in your everyday life? (b) Give an example of when estimating a perimeter is useful. (c) Give an example of when measuring a perimeter is necessary. 88 This file is copyright 07, Rice University. All Rights Reserved.

90 Shape l 89 This file is copyright 07, Rice University. All Rights Reserved.

91 Shape ll 90 This file is copyright 07, Rice University. All Rights Reserved.

92 Shape lll 9 This file is copyright 07, Rice University. All Rights Reserved.

93 Shape lv 9 This file is copyright 07, Rice University. All Rights Reserved.

94 Shape V 9 This file is copyright 07, Rice University. All Rights Reserved.

95 Shape Vl 9 This file is copyright 07, Rice University. All Rights Reserved.

96 Manipulative Mathematics Name Measuring Area and Perimeter Extra Practice Find the area and perimeter of each shaded region, using this square unit measure. as one square ) Estimated area Estimated perimeter Measured area Measured perimeter ) Estimated area Estimated perimeter Measured area Measured perimeter ) Estimated area Estimated perimeter Measured area Measured perimeter ) Estimated area Estimated perimeter Measured area Measured perimeter 5) Estimated area Estimated perimeter Measured area Measured perimeter 95 This file is copyright 07, Rice University. All Rights Reserved.

97 Manipulative Mathematics Coin Lab Name Team Members: To start this activity, put a handful of coins into a bag. The goal is to determine the total amount of money in the bag and describe in detail the method you use. ) How much money is in your bag? ) Describe, in words, the method you used to determine how much money is in your bag. List everything you did, step-by-step, so that someone not in your group could follow your directions. ) Show the calculations you used to determine the total value of the money in your bag. 96 This file is copyright 07, Rice University. All Rights Reserved.

98 Manipulative Mathematics Coin Lab Extra Practice Name Fill in the charts to calculate the total value of each set of coins. ) ) Type of coin Number Value ($) Total value ($) Type of coin Number Value ($) Total value ($) Quarters 0.5 Dimes Nickels 0.05 Pennies 0.0 The total $0.95 value of all the coins ) goes here. ) Type of coin Number Value ($) Total value ($) Type of coin Number Value ($) Total value ($) Dimes 8 Quarters 6 Nickels Pennies 9 5) 6) Type of coin Number Pennies 7 Value ($) Total value ($) Type of coin Number Nickels 5 Value ($) Total value ($) Quarters 9 Pennies Dimes Dimes 8 97 This file is copyright 07, Rice University. All Rights Reserved.

99 7) 8) Type of coin Number Value ($) Total value ($) Type of coin Number Value ($) Total value ($) Pennies 9 Dimes Quarters Nickels 9 Nickels Quarters 6 9) 0) Type of coin Number Value ($) Total value ($) Type of coin Number Value ($) Total value ($) Pennies Pennies 9 Nickels 7 Nickels Dimes Dimes Quarters 5 Quarters 5 For more practice finding the value of a handful of coins go to the website: and choose Count. If you are not completely familiar with U.S. coins, you may find it helpful to have the program display the value of each coin next to its picture. 98 This file is copyright 07, Rice University. All Rights Reserved.

100 Manipulative Mathematics Exploring Slopes of Lines Name The concept of slope has many applications in the real world. The pitch of a roof, the grade of a highway, and a ramp for a wheelchair are some places you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill. We will use geoboards to explore the concept of slope. Using rubber bands to represent lines and the pegs of the geoboards to represent points, we have a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, you ll discover how to find the slope of a line. ) Let s work together to see how to use a geoboard to find the slope of a line. (a) Take your geoboard and a rubber band. Stretch the rubber band between two pegs like this: Doesn t it look like a line? (b) Now stretch the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, like this: Be sure to make a 90º angle around the third peg, so one of the two newly formed lines is vertical and the other side is horizontal. You have made a right triangle! To find the slope of the line count the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run. Slope The slope of a line is m rise run rise measures the vertical change run measures the horizontal change 99 This file is copyright 07, Rice University. All Rights Reserved.

101 (c) On your geoboard, what is the rise? (d) What is the run? (e) What is the slope of the line on your geoboard? m rise run m ) Make another line on your geoboard, and form its right triangle. Draw a picture of your geoboard here: (a) What is the rise? (b) What is the run? (c) What is the slope? ) Make more lines on your geoboard, form the right triangle for each, and count their slopes. Draw the triangles below. (a) Slope = (b) Slope = (c) Slope = ) If the left endpoint of a line is higher than the right endpoint, you have to stretch the rubber band down to make the right triangle. When this happens the rise will be negative because you count down from your starting peg. (a) Do any of your lines in exercise have negative slope? (b) Draw a line with negative slope here and calculate its slope: Slope = 00 This file is copyright 07, Rice University. All Rights Reserved.

102 5) Use a rubber band on your geoboard to make a line with each given slope and draw a picture of it.? (a) Slope = (b) Slope = (c) Slope = (hint: )? 6) Make a horizontal line on your geoboard and draw it here. What is the slope of the horizontal line? 7) Make a vertical line on your geoboard and draw it here. What is the slope of the vertical line? 0 This file is copyright 07, Rice University. All Rights Reserved.

103 Manipulative Mathematics Exploring Slopes of Lines- Extra Practice Name Sketch the rise and the run for the line modeled on each geoboard, then calculate the slope of the line. You may want to use the virtual geoboard online at t_.html?open=activities&hidepanel=true&from =topic_t_.html. ) ) ) ) (a) rise = (a) rise = (a) rise = (a) rise = (b) run = (b) run = (b) run = (b) run = (c) slope = (c) slope = (c) slope = (c) slope = 5) 6) 7) 8) (a) rise = (a) rise = (a) rise = (a) rise = (b) run = (b) run = (b) run = (b) run = (c) slope = (c) slope = (c) slope = (c) slope = 0 This file is copyright 07, Rice University. All Rights Reserved.

104 Draw a line with the given slope. 8 9) slope = 0) slope = ) slope = ) slope = This file is copyright 07, Rice University. All Rights Reserved.

105 Manipulative Mathematics Slope of Line Between Two Points Name ) Start with a geoboard and rubber bands. Stretch one rubber band around the middle row of pegs horizontally and the other rubber band around the middle row of pegs vertically to model the x - axis and the y - axis, like this: You now have a small coordinate system, with 5 x 5 and 5 y 5. Each of the pegs on the geoboard represents a point on the graph. For example, the point, is located at the arrow. ) On your geoboard, make a line between the points, and,. (a) Sketch it on the geoboard below. (b) To find the rise and the run, stretch the rubber band into a right triangle, with one side vertical and the other horizontal. Draw your triangle on the geoboard above. (c) What is the rise? (d) What is the run? rise (e) The slope is m. run m 0 This file is copyright 07, Rice University. All Rights Reserved.