# Using Proportions to Solve Percent Problems I

Size: px
Start display at page:

Transcription

1 RP7-1 Using Proportions to Solve Percent Problems I Pages Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving easy proportions. Prior Knowledge Required: Can write equivalent ratios Can name a ratio from a picture Vocabulary: comparison fraction, equivalent ratios, multiplier, part-to-whole ratio, percent, proportion, ratio Materials: BLM Three Types of Percent Problems (p. N-79) Using pictures to review equivalent ratios. Draw on the board: Have students brainstorm ways of interpreting this picture. SAY: The picture shows four equivalent statements. Write on the board: 6 of the circles are shaded. 9 of the circles are shaded. 6 is of 9 6 : 9 : Exercises: Write four equivalent statements for the picture. a) b) Answers: a) 6/8 are shaded, /4 are shaded, 6 is /4 of 8, 6 : 8 : 4; b) 8/1 are shaded, / are shaded, 8 is / of 1, 8 : 1 : Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-5

2 Writing part-to-whole as a ratio. Tell students that they can write different part-to-whole ratios from the same picture. Draw on the board: ASK: How many circles are there? (9) How many are shaded? () Write on the board: part : whole : 9 ASK: How many groups are there? () How many groups are shaded? (1) Write on the board: part : whole 1 : Exercises: Write a pair of equivalent ratios for the picture. a) b) Answers: a) part : whole 4 : 10 : 5, b) part : whole 9 : 1 : 4. Writing part-to-whole as a fraction. Tell students that they can write different fractions of the part form from the same picture. Draw on the board: whole ASK: How many circles are there? (1) How many are shaded? () Write on the board: part whole 1 Point to the fraction /1 and SAY: The comparison fraction is /1. ASK: How many groups are there? (4) How many groups are shaded? (1) Write on the board: part 1 whole 4 Point to the fraction 1/4 and SAY: The comparison fraction is 1/4. Exercises: 1. Write a pair of equivalent fractions for each picture from the previous set of exercises. Answers: a) part/whole 4/10 /5, b) part/whole 9/1 /4 N-6 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

3 . Determine the part, the whole, and the comparison fraction. Write an equivalent fraction. a) is 1 4 of 1 b) 4 is of 6 c) 6 is of 10 5 Answers: a) part, whole 1, comparison fraction 1/4, part/whole /1 1/4; b) part 4, whole 6, comparison fraction /, part/whole 4/6 /; c) part 6, whole 10, comparison fraction /5, part/whole 6/10 /5 Writing ratios with missing parts. Tell students that in the previous exercises, all four numbers in the equivalent ratios or fractions were given. Usually in questions like those, one number (out of four) is missing. Explain to students that to write a proportion, they have to determine the part, the whole, and what fraction or ratio of the whole the part is, then write them in the correct places. Write on the board: 8 is of what number? SAY: In this question, the part is 8 and the whole is missing. ASK: What is the comparison fraction? (/) What is the ratio of part-to-whole? ( : ) Write on the board: 8 :? :, part 8 whole? Emphasize that writing each number in the correct place is very important, because writing a number in the wrong place leads to a wrong answer. Exercises: Determine the part, the whole, and the comparison fraction. Write the proportion, but replace the missing number with a question mark. a) is 1 of what number? b) 4 is 1 of what number? c) 6 is 5 of what number? d) What number is 4 e) What number is 4 5 of 0? f) What number is 7 of 0? of 1? Answers: a) part, whole?, comparison fraction 1/, so 1/ /?; b) part 4, whole?, comparison fraction 1/, so 1/ 4/?; c) part 6, whole?, comparison fraction /5, so /5 6/?; d) part?, whole 0, comparison fraction /4, so /4?/0; e) part?, whole 0, comparison fraction 4/5, so 4/5?/0; f) part?, whole 1, comparison fraction /7, so /7?/1 Changing a verbal proportion problem into a known problem. Write on the board: 1 is how many fifths of 0? Underline how many fifths and point out that this is the same as?/5. SAY: The denominator tells you that the size of the parts is a fifth, and the numerator the unknown tells you the Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-7

4 number of fifths. So 1 is how many fifths of 0 is another way of saying 1 is?/5 of 0. This is now easy to change to an equivalent ratio. Write on the board:? : 5 1 : 0 Exercises: Write an equivalent ratio for the question. Then write the fraction form. a) 8 is how many thirds of 1? b) 1 is how many quarters of 8? c) 18 is how many tenths of 0? Answers: a)? : 8 : 1,?/ 8/1; b)? : 4 1 : 8,?/4 1/8; c)? : : 0,?/10 18/0 Writing percent statements in terms of ratios. Remind students that asking how many hundredths is like asking for?/100. ASK: What is another name for a fraction with denominator 100? PROMPT: What do we use to compare numbers to 100? (a percent) Since students can write fraction statements as equivalent ratios, and a percent is just a fraction with denominator 100, students can now write percent statements as equivalent ratios. Exercises: Write the proportion (without solving it). Then write the proportion in terms of fractions. Replace the missing number with a question mark. a) 19 is how many hundredths of 0? b) 1 is how many hundredths of 50? c) 6 is how many hundredths of 60? Answers: a)? : : 0,?/100 19/0; b)? : : 50,?/100 1/50; c)? : : 60,?/100 6/60 Remind students that a percent is a hundredth, so asking what is 15% of 40 is asking what is 15 hundredths of 40. If they know how to find a fraction of a whole number, then they know how to find a percent of a whole number. In questions in which the percent is unknown, students can write a comparison fraction with denominator 100 and a question mark in the numerator. Exercises: Write the question as a proportion, in ratio form and in fraction form. a) What is 15% of 40? b) What is % of 50? c) What is 75% of 48? d) 4 is 80% of what number? e) 6 is 5% of what number? f) 1 is 0% of what number? g) What percent of 0 is 19? h) What percent of 4 is 6? Answers: a)? : : 100, or?/40 15/100; b)? : 50 : 100, or?/50 /100; c)? : : 100, or?/48 75/100; d) 4 :? 80 : 100, or 4/? 80/100; e) 6 :? 5 : 100, or 6/? 5/100; f) 1 :? 0 : 100, or 1/? 0/100; g) 19 : 0? : 100, or 19/0?/100; h) 6 : 4? : 100, or 6/4?/100 Distribute BLM Three Types of Percent Problems. All three types of questions from the exercises above are summarized on the BLM. Students can use BLM Three Types of Percent Problems as a reference to help them solve the remaining exercises in this lesson. N-8 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

5 Solving proportions. Show students how to solve proportions using equivalent ratios. Use this problem: If 4 bus tickets cost \$9, how much would 1 tickets cost? Step 1: Make the proportion. Write a fraction on the board, the top of which is the unknown quantity (in this example, Dollars) and the bottom of which is the other quantity (in this example, Tickets). Write on the board the complete ratio of dollars to bus tickets (9 : 4) in fraction form, then write the incomplete ratio of dollars to bus tickets (? : 1) in fraction form, as shown below: Dollars 9? Tickets 4 1 Step : Find the multiplier. Find the number the first denominator is being multiplied by to get the second denominator (in this example, ). Write on the board: 9 4? 1 Step : Find the missing number. Multiply the numerator by that multiplier to find the missing number, as shown below: SAY: Since 9/4 7/1, 1 tickets cost \$7. Have volunteers complete the first few exercises below, then have students answer the rest on their own. Exercises: a) If bus tickets cost \$4, how much will 15 bus tickets cost? b) Five bus tickets cost \$6. How many can you buy with \$0? c) On a map, cm represents 10 km. How many kilometers do 15 cm represent? d) Milly gets paid \$5 for hours of work. How much would she get paid for working 6 hours? e) Three centimeters on a map represents 0 km in real life. If a lake is 6 cm long on the map, what is its actual length? f) There are apples in a bowl for every oranges. If there are 1 oranges, how many apples are there? Bonus: A goalie stopped 18 out of every 19 shots. There were 8 shots. How many goals were scored? Hint: How many did she not stop? Answers: a) \$0, b) 5, c) 50 km, d) \$50, e) 40 km, f) 8, Bonus: Extensions 1. Determine decimals as the value of a percent. a) What percent of 0 is 16.5? b) What percent of 18 is.7? c) What percent of 14 is.8? Answers: a) 55%, b) 15%, c) 0% Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-9

6 . Give word problems involving decimals as the value of a percent. a) A book that costs \$18 came to \$0.70 after taxes. i) How much were the taxes? ii) What percent is the tax? b) The regular price of a book is \$18. The sale price is \$1.60. i) How much was taken off the regular price? ii) What percent was taken off the regular price? Answers: a) i) \$.70, ii) 15%; b) i) \$5.40, ii) 0% N-40 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

7 RP7- Using Proportions to Solve Percent Problems II Pages Standards: 7.RP.A. Goals: Students will write equivalent statements for proportions by keeping track of the part and the whole, and by solving the proportions. Prior Knowledge Required: Can write equivalent ratios Can solve proportions Vocabulary: equivalent ratios, lowest terms, multiplier, percent, proportion, ratio Materials: calculators BLM Three Types of Percent Problems (p. N-79) NOTE: Students can use BLM Three Types of Percent Problems as a reference to help them solve the exercises in this lesson. Review percent proportions in terms of fractions. Remind students that to write a proportion, they have to determine the part, the whole, and what fraction of the whole is the part. SAY: Suppose that we want to find what percent of 5 is 7. ASK: What is the whole in this question? (5) What is the part? (7) ASK: What is the part-to-whole fraction? (7/5) SAY: There is another way of writing the part-to-whole, which is the missing percent/100 or?/100, so we can equate the two part-to-whole ratios. Write on the board: part 7? whole Emphasize that writing each number in the correct place is very important, because writing a number in the wrong place leads to a wrong answer. Remind students that writing part-to-whole ratios is the first step of solving proportions. In the second step, they have to find the relation between two given numerators or denominators. In this example, students have to find the number the first denominator is being multiplied by to get the second denominator. Write on the board: 7 5 4? 100 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-41

8 In the third step, students have to multiply the numerator by that multiplier to find the missing number, as shown below: Exercises: Write the proportion in fraction form, then solve the proportion. a) What percent of 0 is 9? b) What is 50% of 50? c) 9 is what percent of 5? d) 1 is 6% of what number? Answers: a) 9/0?/100, so 9 is 45% of 0; b)?/50 50/100, so 5 is 50% of 50; c) 9/5?/100, so 9 is 6% of 5; d) 1/? 6/100, so 1 is 6% of 50 (MP.1) Solving proportions that need simplifying. Write on the board: 9? Explain to students this proportion is easy to solve because the relationship between the two denominators is obvious. SAY: You can find the second denominator by multiplying the first denominator by 5. You find the multiplier by dividing 100 by 0. Write on the board: 7? SAY: In this proportion, the relation between two denominators is not clear. Ask students to use their calculators to divide 100 by 5 to find the multiplier. ASK: Is the answer a well-known decimal? (no) SAY: Don t give up! Try to reduce 7/5 to lowest terms. Write on the board: SAY: Replace 7/5 by 1/5. Write on the board: 1? ASK: Is this proportion easy to solve? (yes) Ask a volunteer to solve the proportion as shown below: , so N-4 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

9 Exercises: Find an equivalent ratio to rewrite the proportion. Solve the new proportion.? 11 6? 1 40 a) b) c) ? 100? 4? 75 d) e) f) 1? Answers: a)?/100 1/, so? 50; b) 1/4?/100, so? 5; c) 1/? /5, so? 0; d)?/100 /4, so? 75; e)?/48 /4, so? 6; f) 1/5?/100, so? 0 Word problem practice. Exercise: Use your answer to each problem to obtain the answer to the next problem. Discuss the similarities and differences between the problems. a) 1 is how many fifths of 0? b) How many fifths of 0 is 1? c) 1 is what percent of 0? d) What percent of 0 is 1? e) A shirt costs \$0, and \$1 was taken off. What percent was taken off? Answers: a), b), c) 40, d) 40, e) 40 Selected solutions: a) 1/0?/5, so? ; b)?/5 1/0, so? ; c) 1/0?/100, so? 40. The difference between a) and b) is just the order of fractions, but in c) the question asks for percentage so the denominator is 100. Finding the whole from the part. Write on the board: of a number is 100. What is the number? ASK: Is 100 the part or the whole? (the part) What is the whole? (the number that we don t know) Tell students that this is a part-to-whole ratio. Write on the board: 100 part? whole Have students solve the proportion. (? 150) Exercises: Write the proportion, then find the number. a) of a number is 9 4 b) 4 9 of a number is 4 c) 7 of a number is 1 1 Answers: a) /4 9/?, so the number is 1; b) 4/9 4/?, so the number is 54; c) 7/1 1/?, so the number is 9 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-4

13 improper fraction. We cannot write 1,00 ( 1/) 1,00 ( + 1/) 1,00 + 1,00 1/, because the distributive law does not apply here. If you try to use the distribute law, you should see right away that doing so gives an answer that can t be correct: 1,00 is larger than what you started with, 1,00 ( 1/), because is less than 1/ and dividing by a smaller number gives a larger result of the people (boys, girls, adults) at the park are boys. There are more girls than boys. There are 7 adults. How many people are at the park? Solution: boys girls adults From the model, it is clear that 1/5 of the total number of people is 10. 1/5 of 50 is 10, so there are 50 people at the park. Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-47

14 RP7- Solving Equations (Introduction) Pages 51 5 Standards: preparation for 7.EE.B.4 Goals: Students will use the balance model to solve addition and multiplication equations including negative addends and coefficients. Prior Knowledge Required: Is familiar with balances Can solve a simple equation to find an unknown value Can substitute numbers for unknowns in an expression Can check whether a number solves an equation Vocabulary: balance, equation, expression, integer, pan balance, quotient, sides (of an equation), variable Materials: pan balance apples, cubes, or other small objects for demonstration and for each pair of students to use with a pan balance paper bags, one per pair of students and some for display masking tape 40 connecting cubes for demonstrations NOTE: You will need a number of identical objects for demonstrations throughout this lesson. The objects you use should be significantly heavier than a paper bag, so that the presence of a paper bag on one of the pans of the balance does not skew the pans. Apples are used in the lesson plan below (to match the pictures in the AP Book), but other objects, like small fruit of equal size, metal spoons, golf balls, tennis balls, or cereal bars, will work well. If a pan balance is not available, refer to a concrete model, such as a seesaw, to explain how a pan balance works, and use pictures or other concrete models during the lesson. Review pan balances. Show students a pan balance. Place the same number of identical (or nearly identical) apples on both pans, and show that the pans balance. Remind students that when the pans, or scales, are balanced, this means there is the same number of apples on each pan. Removing the same number of apples from both pans keeps them balanced. Place some apples in a paper bag and place it on one pan, then add some apples beside the bag. Place the same total number of apples on the other pan. ASK: Are the pans balanced? (yes) What does this mean? (the same number of apples are on each pan) Take one apple off each pan. ASK: Are the pans still balanced? Repeat with two apples. Remove the same number of apples N-48 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

16 ASK: How many apples are left on the right side of the equation? (5) What letter did we use to represent how many apples are in the bag? (x) Remind students that we write this as x 5. Solving addition equations without using the balance model. Present a few equations without a corresponding model. Have students signal how many apples need to be subtracted from both sides of the equation, then write the vertical subtraction for both sides. Exercises: Solve the equation. a) x b) n + 17 c) 14 + n 17 d) p Answers: a) x 4, b) n 6, c) n, d) p 6 Students who have trouble deciding how many apples to subtract without the model can complete the following problems. Exercises: Write the missing number. Part a) has been done for you. a) x + 15 b) x + 55 c) x + 91 Bonus: x x x x 8 Answers: b) 55, c) 91, Bonus: x Finally, give students a few more equations and have them work through the whole process of subtracting the same number from both sides to find the unknown number. Exercises: Solve the equation by subtracting the same number from both sides to find the unknown number. a) x b) x c) + x 5 d) x Sample solution: a) x x 9 Bonus: The scale below is balanced. Each bag has the same number of apples in it. How many apples are in the bag? Hint: You can cross out whole bags too! Answers: b) 1, c), d) 6, Bonus: Solving multiplication equations given by a model. Divide a desk into two parts using masking tape and place three bags (with 4 cubes in each) on one side of the line, and 1 separate cubes on the other side of the line. Tell students that the pans are balanced. ASK: What does this say about the number of cubes in both pans? (they are equal) How many cubes are on the pan without the bags? (1) How many cubes are in the bags in N-50 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

18 Multiplying and dividing by the same number does not change the starting number. Write on the board: (5 ) ( ) (8 ) (5 4) 4 (9 ) (10 6) 6 Have students solve each question. (5,, 8, 5, 9, 10) SAY: Look at the questions you solved. ASK: How are they all the same? (they start with a number, multiply by another number, then divide by the same number) Did you get back to the same number you started with? (yes) Does it matter what number you started with? Does it matter what number you multiplied and divided by as long as it was the same number? (no) Have students write their own question of the same type, and check that they get back to the same number they started with. Explain that you can do the same with unknown numbers. Show one paper bag with some cubes and SAY: I want to multiply this by. ASK: What will the answer look like? ( bags) SAY: I want to divide the result by. What will you get? (1 bag again) Write on the board: ( ) ASK: What will we get when we perform the multiplication and the division? (the box) Write on the board: (b ) (b 5) 5 (b 6) 6 (b 10) 10 Have students solve each question. (, 5, 6, 10) Solving equations by dividing both sides by the same number. Write on the board: (b 7) b (b ) b (b 4) b (b 8) b (b 1) b (b 9) b For each equation, have students hold up the correct number of fingers to signal the number they would divide the product by to get back to b. (7,, 4, 8, 1, 9) If a pan balance is available, show students the balance with bags of 5 apples (other objects will work equally well cereal bars, metal spoons, etc.) on one pan, and 15 apples on the other pan. Invite a volunteer to write the equation for the balance on the board, as shown below: 5 15 ASK: How many apples are in one bag? (5) Have a volunteer make three groups of five apples on the side without the bags. Point out that there are three equal groups of apples on both sides of the balance. Remove two of the bags from one side, and two of the groups from the other side. SAY: I have replaced three equal groups on each side with only one of these groups. N-5 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

21 Extensions (MP.5) a) Does the model work to solve the equation? i) x + 8 ii) x 8 iii) x 1 iv) x 1 v) 0.6x 1.8 b) Does doing the same thing to both sides work? Answers: a) i) yes, ii) no, iii) yes, iv) no, v) no; b) i) yes, ii) yes, iii) yes, iv) yes, v) yes Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-55

22 RP7-4 Cross-Multiplication (Introduction) Pages Standards: 7.RP.A. Goals: Students will cross-multiply to write an equation for problems involving proportions. Prior Knowledge Required: Can convert a fraction a/b to a decimal by dividing a b Can find equivalent fractions by multiplying the numerator and denominator by the same number Can write an equivalent multiplication statement for a given division statement Vocabulary: canceling, commutative property, complex fraction, cross-multiply, equivalent fractions Materials: calculators Review writing a fraction as a division statement. Remind students that we can calculate the value of a fraction such as /4 by dividing 4. For a quick reminder of why this is true, SAY: To find 1/4 of something, I would divide it into four equal groups. So to find 1/4 of something, divide it by 4. You can think of 1/4 as 1/4 of 1, so that is 1 4. But /4 is three times as much as 1/4, so /4 is Exercises: Write as a division statement and use a calculator to find the answer. a) b) 5 c) 7 d) e) Answers: a) 5 0.6, b) , c) , d) 10 0., e) Writing fraction statements as equivalent multiplication statements. Remind students that a division statement can be written as a multiplication statement. For example, 1 4 can be rewritten as 1 4. Exercises: Change the division statements in the previous set of exercises to multiplication statements. Answers: a) 5 0.6, b) , c) , d) 10 0., e) To guide students in the following exercises, write this template on the board: so N-56 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

23 Exercises: Change the fraction statement to a division statement, then to a multiplication statement. a) b) c) d) Answers: a) , so ; b) , so ; c) , so ; d) ; so Finding a pattern. Now have students look at their answers to the questions in the previous set of exercises. ASK: If we know the value of a fraction as a decimal, how can we use that to write a multiplication statement? Write on the board: ASK: In which blank does the numerator the top number of the fraction go? (the first blank) What number goes in the second blank, the denominator or the value? (it doesn t matter, because multiplication follows the commutative property) Explain that when you know the decimal value of a fraction, the numerator of the fraction can be written as the product of the denominator and the decimal value. Exercises: 1. Write the fraction as a product. a) b) c) d) Answers: a) , b) , c) , d) Calculate the value of the fraction, then write a multiplication statement. a) b) 9 c) 1 d) Bonus: e) f) Answers: a) 0.4 5, b) , c) , d) , Bonus: e) 4.6 5, f) Writing fraction statements that involve variables as a product. Write on the board: 10 x SAY: I don t know what number x is, but I know that whatever it is, times x is equal to 10. Write on the board: 10 x, so x 10 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-57

24 Exercises: Rewrite the equation so that it uses multiplication instead of division. a) 4 x b) 4 x c) 4 4 x d) x 5 x x e) 5 f) g) 8 x h) 15 x i) 18 x j) 18 x k) x x l) 15 Answers: a) 4 x, b) 4 x, c) 4 4x, d) x 5, e) x 5 6, f) x 8 7, g) 8 x, h) 15 x, i) 18 x, j) 18 x, k) x, l) x 15 Changing an equation of equivalent fractions to an equation of multiplication statements. Write on the board: SAY: I can write each fraction as a division statement. Write on the board: Have students verify this equation by doing long division. ( and ) Tell students that you find it easier to work with multiplication than with division. SAY: I would like to be able to verify this equality by using multiplication instead of division, and I know a trick that lets me change the equation so I can do that. Work through the steps below as a class. Write on the board: SAY: Start by multiplying both sides by 5 0 (the product of the denominators). Write on the board: SAY: Then, cancel common factors and rewrite the equation. The equations should look like this: Point out that we have now created an equation of multiplication statements instead of fractions. Have students use multiplication to verify the equation. ASK: Was it easier to use multiplication to verify the equation or was it easier to use division? (multiplication) Have students use this method to complete the following exercises. N-58 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

25 Exercises: Change the equivalent fractions to equivalent multiplication statements. a) 8 b) 6 c) 5 10 d) Answers: a) 1 8, b) , c) , d) Multiplying to verify equivalent fractions. Point out that the fractions /5 and 1/0 are equivalent fractions. ASK: How do I know? PROMPT: What number can we multiply both the numerator and the denominator by in /5 to get 1/0? (multiply by 4 to get 1 and 5 by 4 to get 0) Write on the board: ASK: How can we change this to an equation with multiplication instead? (we did it above it was 0 1 5) Exercises: Change the equivalent fractions to equivalent multiplication statements. a) b) c) 7 1 d) Sample solution: a) /4 15/0 / / Answers: b) , c) , d) (MP.8) Finding a pattern (cross-multiplying). Have students look at their answers to the previous set of exercises. ASK: How can you find which numbers to multiply together from the fractions? PROMPT: Do you multiply both numerators together? (no) What do you multiply together? (the numerator of one fraction with the denominator of the other fraction) Go through each one, point to the answer, and verify that this is indeed what students did for each question join the numerator of each fraction with the denominator of the other fraction to emphasize this point. Tell students that because the products from equivalent fractions can be found by drawing an X, we call this process cross-multiplying. Write on the board: Exercises: 1. Verify that each pair of fractions in the previous two sets of exercises are in fact equivalent by verifying that the products you found are equal. Answers: a) 1 8 b) c) d) a) b) c) d) Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-59

26 (MP.7). Use cross-multiplication to verify that the fractions are equivalent. a) 6 b) 4 1 c) Bonus: Answers: a) 14 4 and 6 7 4; b) and ; c) and ; Bonus: ,19 and 1 4,19 (MP.7) Cross-multiply to identify equivalent fractions. For the following exercises, have students decide whether the two fractions are equivalent by multiplying the numerator of each fraction with the denominator of the other fraction and checking whether the two products are equal. Exercises: Cross-multiply to check if the fractions are equivalent. a) 8 64 and b) 7 49 and c) 6 and d) 4 6 and e) and f) and g) 7 6 and Bonus: and Answers: a) not equivalent, b) equivalent, c) not equivalent, d) not equivalent, e) not equivalent, f) equivalent, g) equivalent, Bonus: not equivalent Cross-multiplying for complex fractions. SAY: You can cross-multiply complex fractions, too. Explain that complex fractions are like other fractions they just contain fractions in the numerator, or the denominator, or both. Write on the board: 4 5 and SAY: To verify that they are equivalent, I have to multiply the numerator of the first complex fraction by the denominator of the second complex fraction, then the numerator of the second complex fraction by the denominator of the first complex fraction. Write on the board: and ASK: Are they equal? (yes) Students can answer by signaling thumbs up. ASK: How do you know? (because 4/1 and 1/ are equivalent fractions) SAY: So the two complex fractions are equivalent. Exercises: Cross-multiply to check if the complex fractions are equivalent. 6 1 a) 4 and 4 b) 5 and Answers: a) equivalent, b) not equivalent N-60 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

27 Explain that when the relation between two numerators or two denominators is clear and easy to find, they can find the missing number mentally. Otherwise, it is better to use cross-multiplication. Exercises: How would you solve the proportion: mentally or using cross-multiplication? Circle the questions you would solve mentally. a) 6 b) 5? c) 5? ? d) 55? e) f) ? 65? Answers: a) mentally,? 10; b) cross-multiplying,? 1.5; c) cross-multiplying,?.5; d) mentally,? 11; e) mentally,? 7.7; f) cross-multiplying,?.5 Extensions 1. a) Have students investigate this question: If fractions 5 and 6 10 is equivalent to? Write on the board: 6 are equivalent, what fraction 6 Cross-multiply to get Have students decide what fractions they can cross-multiply to get Suggest that students look for where the parts of each fraction go in the equation and compare how the equations are different. PROMPT: Which numbers switched positions, and which numbers stayed in the same position? b) Have students cross-multiply to make a new pair of equivalent fractions, then use the commutative property of multiplication for one of the products (not both!) to make a new pair of equivalent fractions. i) 6, so ii) 1 5, so iii) 9, so Answers: a) and 10 are in the same position, but 5 and 6 get switched. So the corresponding fractions are 5 ; b) i) /6 /9, ii) 1/5 4/0, iii) /9 5/15. Emphasize to students that to 6 10 find the second pair of equivalent fractions, they can read the numbers from the first pair across, from left to right.. Mental math and estimation. Tell students that you know someone who changed the fractions in Extension 1, part b.ii) to 1/0 5/4. ASK: How can you tell immediately that this is wrong? Answer: 1/0 is less than 1, but 5/4 is more than 1. Cross-multiplying with decimal numbers. Cross-multiply to verify whether the fractions are equivalent. a) and b) and Sample solution: a) , Answers: b) no, c) no c) and Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-61

28 RP7-5 Using Equations to Solve Proportions Pages Standards: 7.RP.A. Goals: Students will cross-multiply to solve problems that involve proportions. Prior Knowledge Required: Can convert a fraction a/b to a decimal by dividing a b Can find equivalent fractions by multiplying the numerator and denominator by the same number Can write a proportion to solve ratio and percent problems Can solve multiplicative equations Can write an equivalent multiplication statement for a given division statement Vocabulary: cross-multiply, equation, equivalent fractions, equivalent ratios, percent, proportion, variable Materials: calculators Using cross-multiplying to write equations. Show students how to cross-multiply to write an equation when there is a variable in one of the fractions. Write on the board: 10 x SAY: I don t know what number x is, but I know that no matter what, times x is equal to 10 times. Write on the board: x, so 0 x x Exercises: Cross-multiply to write an equation for x. a) 4 x 5 b) 4 x 5 c) 4 4 x 5 d) x 1 9 x 5 x 8 8 x e) f) g) h) 15 x Answers: a) 4 5 x, so 10 x; b) 10 x, c) 10 4x, d) 9x 6, e) x 0, f) 8x 56, g) 40 x, h) 60 x Using cross-multiplying to solve equations. Review multiplicative equations like b 1. Remind students that to solve this type of equation, they have to divide both sides of the equation by the coefficient of the unknown. For example, in the equation b 1, the answer is b 1, so b 6. N-6 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

29 Exercises: 1. Have students solve the equations in their answers to the previous set of exercises. Sample solution: d) 9x 1, 9x 6, x 6 9, x 4 Answers: a) x 60, b) x 40, c) x 0, e) x 10, f) x, g) x 0, h) x 0. Rewrite the equation so that it involves multiplication, then solve for x. Check your answer by substitution. a) 0 5 x b) x c) x d) x e) x f) x 8 Answers: a) 0 5x, so x 0 5 4; b) x 6 7 4; c) x 6, so x 6 1; d) 60 15x, so x ; e) x 7 9 6; f) 48 8x, so x Point out that parts c) and f) do not even need to be rewritten, as they can be solved in one step. For example, c) says directly that 6 x, so we don t need to first write that x 6. Cross-multiplying when the answers are decimal numbers. Tell students to again crossmultiply to solve for x, but this time their answers will be decimal numbers. This means that they are comparing equivalent ratios rather than equivalent fractions. Remind students that we can write ratios in fraction form even when both terms are not whole numbers. Review writing fractions as decimal fractions, then as decimals. Write on the board: , 1 0.5, , 4 0.4, 6 0.6, Exercises: Solve for x. a) 10 x 5 7 x b) c) d) 7 4 x x 5 e) 9 6 x 5 f) 5 11 x Sample solution: c) 5x 4, so x 4 5 4/5 8 /5, so x 8.4 Answers: a) 10x 9, so x 0.9; b) 6x 15, so x.5; d) 4x 5, so x 8.75; e) 6x 45, so x 7.5; f) 5x, so x 6.6 Using proportions to solve percent problems. Review writing a proportion to solve a percent problem, then demonstrate how using cross-multiplication makes the problem easy. Write on the board: What is 0% of 8? SAY: Suppose the answer is x and we re going to find x. If 0% of 8 is equal to x, the ratio of x to 8 is the same as the ratio of 0 to 100. Write on the board: x : 8 0 : 100 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships N-6

30 SAY: To solve this proportion, you can write it as two equivalent fractions. Write on the board: x : 8 0 : 100 x Remind students that they already used cross-multiplying to solve this type of equation. Ask a volunteer to solve the equation, as shown below. x 0, so 100x x 40 x x.4 Explain to students that writing the proportion is the most important part of the solving process. Write on the board: 1 is % of what number? SAY: Suppose the answer is x and we re going to find x. Have students propose different ways of writing the proportion for this problem. SAY: Because there is % in the question, one fraction in the proportion is /100. Write on the board: x x 100 ASK: Which proportion gives me the answer, the first or the second? (the second) Students can answer by signaling thumbs up or thumbs down as you point to each proportion. ASK: How do you know? (because 1 is a part of the question and is in the numerator of the second proportion) Ask a volunteer to use cross-multiplying to solve the proportion, as shown below. 1, so x x 100 x 1,00 x 1,00 x 400 Exercises: Write a proportion in fraction form, then cross-multiply and solve. a) What is 15% of 40? b) What is % of 50? c) What is 75% of 48? d) 4 is 80% of what number? e) 6 is 5% of what number? f) 1 is 0% of what number? Answers: a) x/40 15/100, x 6; b) x/50 /100, x 16; c) x/48 75/100, x 6; d) 4/x 80/100, x 0; e) 6/x 5/100, x 48; f) 1/x 0/100, x 40 N-64 Teacher s Guide for AP Book 7. Unit Ratios and Proportional Relationships

### EE6-5 Solving Equations with Balances Pages 77 78

EE6-5 Solving Equations with Balances Pages 77 78 STANDARDS 6.EE.B.5, 6.EE.B.6 Goals Students will use pictures to model and solve equations. Vocabulary balance equation expression sides (of an equation)

### OA3-10 Patterns in Addition Tables

OA3-10 Patterns in Addition Tables Pages 60 63 Standards: 3.OA.D.9 Goals: Students will identify and describe various patterns in addition tables. Prior Knowledge Required: Can add two numbers within 20

### Unit 7 The Number System: Multiplying and Dividing Integers

Unit 7 The Number System: Multiplying and Dividing Integers Introduction In this unit, students will multiply and divide integers, and multiply positive and negative fractions by integers. Students will

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Unit 6 Number and Operations in Base Ten: Decimals

Unit 6 Number and Operations in Base Ten: Decimals Introduction Students will extend the place value system to decimals. They will apply their understanding of models for decimals and decimal notation,

### NS6-50 Dividing Whole Numbers by Unit Fractions Pages 16 17

NS6-0 Dividing Whole Numbers by Unit Fractions Pages 6 STANDARDS 6.NS.A. Goals Students will divide whole numbers by unit fractions. Vocabulary division fraction unit fraction whole number PRIOR KNOWLEDGE

### NF5-12 Flexibility with Equivalent Fractions and Pages 110 112

NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.

### Five Ways to Solve Proportion Problems

Five Ways to Solve Proportion Problems Understanding ratios and using proportional thinking is the most important set of math concepts we teach in middle school. Ratios grow out of fractions and lead into

### MD5-26 Stacking Blocks Pages 115 116

MD5-26 Stacking Blocks Pages 115 116 STANDARDS 5.MD.C.4 Goals Students will find the number of cubes in a rectangular stack and develop the formula length width height for the number of cubes in a stack.

### Unit 2 Number and Operations Fractions: Multiplying and Dividing Fractions

Unit Number and Operations Fractions: Multiplying and Dividing Fractions Introduction In this unit, students will divide whole numbers and interpret the answer as a fraction instead of with a remainder.

### Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

### OA4-13 Rounding on a Number Line Pages 80 81

OA4-13 Rounding on a Number Line Pages 80 81 STANDARDS 3.NBT.A.1, 4.NBT.A.3 Goals Students will round to the closest ten, except when the number is exactly halfway between a multiple of ten. PRIOR KNOWLEDGE

### 10-4-10 Year 9 mathematics: holiday revision. 2 How many nines are there in fifty-four?

DAY 1 Mental questions 1 Multiply seven by seven. 49 2 How many nines are there in fifty-four? 54 9 = 6 6 3 What number should you add to negative three to get the answer five? 8 4 Add two point five to

### Fractions, decimals and percentages

Fractions, decimals and percentages Some notes for the lesson. Extra practice questions available. A. Quick quiz on units Some of the exam questions will have units in them, and you may have to convert

### Chapter 1: Order of Operations, Fractions & Percents

HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Assessment For The California Mathematics Standards Grade 6

Introduction: Summary of Goals GRADE SIX By the end of grade six, students have mastered the four arithmetic operations with whole numbers, positive fractions, positive decimals, and positive and negative

### Rational Number Project

Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson : Overview Students estimate sums and differences using mental images of the 0 x 0 grid. Students develop strategies for adding

### The Distributive Property

The Distributive Property Objectives To recognize the general patterns used to write the distributive property; and to mentally compute products using distributive strategies. www.everydaymathonline.com

### Pre-Algebra Lecture 6

Pre-Algebra Lecture 6 Today we will discuss Decimals and Percentages. Outline: 1. Decimals 2. Ordering Decimals 3. Rounding Decimals 4. Adding and subtracting Decimals 5. Multiplying and Dividing Decimals

### Scope and Sequence KA KB 1A 1B 2A 2B 3A 3B 4A 4B 5A 5B 6A 6B

Scope and Sequence Earlybird Kindergarten, Standards Edition Primary Mathematics, Standards Edition Copyright 2008 [SingaporeMath.com Inc.] The check mark indicates where the topic is first introduced

### BASIC MATHEMATICS. WORKBOOK Volume 2

BASIC MATHEMATICS WORKBOOK Volume 2 2006 Veronique Lankar A r ef resher o n t he i mp o rt a nt s ki l l s y o u l l ne e d b efo r e y o u ca n s t a rt Alg e b ra. This can be use d a s a s elf-teaching

### Solving Rational Equations

Lesson M Lesson : Student Outcomes Students solve rational equations, monitoring for the creation of extraneous solutions. Lesson Notes In the preceding lessons, students learned to add, subtract, multiply,

### Ratio and Proportion Study Guide 12

Ratio and Proportion Study Guide 12 Ratio: A ratio is a comparison of the relationship between two quantities or categories of things. For example, a ratio might be used to compare the number of girls

### Math Refresher. Book #2. Workers Opportunities Resources Knowledge

Math Refresher Book #2 Workers Opportunities Resources Knowledge Contents Introduction...1 Basic Math Concepts...2 1. Fractions...2 2. Decimals...11 3. Percentages...15 4. Ratios...17 Sample Questions...18

### Exponents. Exponents tell us how many times to multiply a base number by itself.

Exponents Exponents tell us how many times to multiply a base number by itself. Exponential form: 5 4 exponent base number Expanded form: 5 5 5 5 25 5 5 125 5 625 To use a calculator: put in the base number,

### Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

### Welcome to Basic Math Skills!

Basic Math Skills Welcome to Basic Math Skills! Most students find the math sections to be the most difficult. Basic Math Skills was designed to give you a refresher on the basics of math. There are lots

### Fractions. Chapter 3. 3.1 Understanding fractions

Chapter Fractions This chapter will show you how to find equivalent fractions and write a fraction in its simplest form put fractions in order of size find a fraction of a quantity use improper fractions

### Paramedic Program Pre-Admission Mathematics Test Study Guide

Paramedic Program Pre-Admission Mathematics Test Study Guide 05/13 1 Table of Contents Page 1 Page 2 Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Page 9 Page 10 Page 11 Page 12 Page 13 Page 14 Page 15 Page

### MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

### Oral and mental starter

Lesson Objectives Order fractions and position them on a number line (Y6) Vocabulary gauge, litre numerator, denominator order Resources OHT. individual whiteboards (optional) Using fractions Oral and

### Fractions. If the top and bottom numbers of a fraction are the same then you have a whole one.

What do fractions mean? Fractions Academic Skills Advice Look at the bottom of the fraction first this tells you how many pieces the shape (or number) has been cut into. Then look at the top of the fraction

### Fraction Problems. Figure 1: Five Rectangular Plots of Land

Fraction Problems 1. Anna says that the dark blocks pictured below can t represent 1 because there are 6 dark blocks and 6 is more than 1 but 1 is supposed to be less than 1. What must Anna learn about

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### How do you compare numbers? On a number line, larger numbers are to the right and smaller numbers are to the left.

The verbal answers to all of the following questions should be memorized before completion of pre-algebra. Answers that are not memorized will hinder your ability to succeed in algebra 1. Number Basics

### Multiplication. Year 1 multiply with concrete objects, arrays and pictorial representations

Year 1 multiply with concrete objects, arrays and pictorial representations Children will experience equal groups of objects and will count in 2s and 10s and begin to count in 5s. They will work on practical

### Dr Brian Beaudrie pg. 1

Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, 3 0.14 means add 0.14 three times, regroup, and simplify,

### The Crescent Primary School Calculation Policy

The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has

### Preliminary Mathematics

Preliminary Mathematics The purpose of this document is to provide you with a refresher over some topics that will be essential for what we do in this class. We will begin with fractions, decimals, and

### Sample Fraction Addition and Subtraction Concepts Activities 1 3

Sample Fraction Addition and Subtraction Concepts Activities 1 3 College- and Career-Ready Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### Five daily lessons. Page 23. Page 25. Page 29. Pages 31

Unit 4 Fractions and decimals Five daily lessons Year 5 Spring term Unit Objectives Year 5 Order a set of fractions, such as 2, 2¾, 1¾, 1½, and position them on a number line. Relate fractions to division

### All the examples in this worksheet and all the answers to questions are available as answer sheets or videos.

BIRKBECK MATHS SUPPORT www.mathsupport.wordpress.com Numbers 3 In this section we will look at - improper fractions and mixed fractions - multiplying and dividing fractions - what decimals mean and exponents

### Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations

Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)

### Integers, I, is a set of numbers that include positive and negative numbers and zero.

Grade 9 Math Unit 3: Rational Numbers Section 3.1: What is a Rational Number? Integers, I, is a set of numbers that include positive and negative numbers and zero. Imagine a number line These numbers are

### Mental Computation Activities

Show Your Thinking Mental Computation Activities Tens rods and unit cubes from sets of base-ten blocks (or use other concrete models for tenths, such as fraction strips and fraction circles) Initially,

### 5 Mathematics Curriculum

New York State Common Core 5 Mathematics Curriculum G R A D E GRADE 5 MODULE 1 Topic B Decimal Fractions and Place Value Patterns 5.NBT.3 Focus Standard: 5.NBT.3 Read, write, and compare decimals to thousandths.

### MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

### Algebra Unit Plans. Grade 7. April 2012. Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto

Algebra Unit Plans Grade 7 April 2012 Created By: Danielle Brown; Rosanna Gaudio; Lori Marano; Melissa Pino; Beth Orlando & Sherri Viotto Unit Planning Sheet for Algebra Big Ideas for Algebra (Dr. Small)

### CAHSEE on Target UC Davis, School and University Partnerships

UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 006 Director Sarah R. Martinez,

### Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman

Introduce Decimals with an Art Project Criteria Charts, Rubrics, Standards By Susan Ferdman hundredths tenths ones tens Decimal Art An Introduction to Decimals Directions: Part 1: Coloring Have children

### 1 ENGAGE. 2 TEACH and TALK GO. Round to the Nearest Ten or Hundred

Lesson 1.2 c Round to the Nearest Ten or Hundred Common Core Standard CC.3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100. Lesson Objective Round 2- and 3-digit numbers

### Maths Workshop for Parents 2. Fractions and Algebra

Maths Workshop for Parents 2 Fractions and Algebra What is a fraction? A fraction is a part of a whole. There are two numbers to every fraction: 2 7 Numerator Denominator 2 7 This is a proper (or common)

### Calculation Policy Fractions

Calculation Policy Fractions This policy is to be used in conjunction with the calculation policy to enable children to become fluent in fractions and ready to calculate them by Year 5. It has been devised

### 4 Mathematics Curriculum

New York State Common Core 4 Mathematics Curriculum G R A D E GRADE 4 MODULE 1 Topic F Addition and Subtraction Word Problems 4.OA.3, 4.NBT.1, 4.NBT.2, 4.NBT.4 Focus Standard: 4.OA.3 Solve multistep word

### Accentuate the Negative: Homework Examples from ACE

Accentuate the Negative: Homework Examples from ACE Investigation 1: Extending the Number System, ACE #6, 7, 12-15, 47, 49-52 Investigation 2: Adding and Subtracting Rational Numbers, ACE 18-22, 38(a),

### CALCULATIONS. Understand the operation of addition and the associated vocabulary, and its relationship to subtraction

CALCULATIONS Pupils should be taught to: Understand the operation of addition and the associated vocabulary, and its relationship to subtraction As outcomes, Year 4 pupils should, for example: Use, read

Ohio Standards Connection Patterns, Functions and Algebra Benchmark E Solve open sentences and explain strategies. Indicator 4 Solve open sentences by representing an expression in more than one way using

### Decimals and other fractions

Chapter 2 Decimals and other fractions How to deal with the bits and pieces When drugs come from the manufacturer they are in doses to suit most adult patients. However, many of your patients will be very

### Decimal Notations for Fractions Number and Operations Fractions /4.NF

Decimal Notations for Fractions Number and Operations Fractions /4.NF Domain: Cluster: Standard: 4.NF Number and Operations Fractions Understand decimal notation for fractions, and compare decimal fractions.

Ohio Standards Connection: Number, Number Sense and Operations Standard Benchmark B Use models and pictures to relate concepts of ratio, proportion and percent. Indicator 1 Use models and visual representation

### Lesson one. Proportions in the Port of Long Beach 1. Terminal Objective. Lesson 1

Proportions in the Port of Long Beach Lesson one Terminal Objective Content Standard Reference: Students will solve Port of Long Beach word problems by writing a proportion and using the cross product

### Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

CONTENTS Introduction...iv. Number Systems... 2. Algebraic Expressions.... Factorising...24 4. Solving Linear Equations...8. Solving Quadratic Equations...0 6. Simultaneous Equations.... Long Division

### WSMA Decimal Numbers Lesson 4

Thousands Hundreds Tens Ones Decimal Tenths Hundredths Thousandths WSMA Decimal Numbers Lesson 4 Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions

### Free Pre-Algebra Lesson 55! page 1

Free Pre-Algebra Lesson 55! page 1 Lesson 55 Perimeter Problems with Related Variables Take your skill at word problems to a new level in this section. All the problems are the same type, so that you can

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

.2 Methods of Addition Objectives Relate addition stories to number bonds. Write two addition facts for a given number bond. Solve picture problems using addition. Learn addition facts through, and the

### Charlesworth School Year Group Maths Targets

Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve

### Math 0306 Final Exam Review

Math 006 Final Exam Review Problem Section Answers Whole Numbers 1. According to the 1990 census, the population of Nebraska is 1,8,8, the population of Nevada is 1,01,8, the population of New Hampshire

### Session 7 Fractions and Decimals

Key Terms in This Session Session 7 Fractions and Decimals Previously Introduced prime number rational numbers New in This Session period repeating decimal terminating decimal Introduction In this session,

### Math and FUNDRAISING. Ex. 73, p. 111 1.3 0. 7

Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1

### Sunny Hills Math Club Decimal Numbers Lesson 4

Are you tired of finding common denominators to add fractions? Are you tired of converting mixed fractions into improper fractions, just to multiply and convert them back? Are you tired of reducing fractions

### INTRODUCTION TO FRACTIONS

Tallahassee Community College 16 INTRODUCTION TO FRACTIONS Figure A (Use for 1 5) 1. How many parts are there in this circle?. How many parts of the circle are shaded?. What fractional part of the circle

### Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

What five coins add up to a nickel? five pennies (1 + 1 + 1 + 1 + 1 = 5) Which is longest: a foot, a yard or an inch? a yard (3 feet = 1 yard; 12 inches = 1 foot) What do you call the answer to a multiplication

### From the Webisode: Math Meets Fashion

lesson CCSS CONNECTIONS Percent Markups From the Webisode: Math Meets Fashion In this lesson, s solve a multi-step problem by identifying percent markups of a whole and calculating a final sale price.

### Planning For Success Mathematics: Numeration Inquiry Investigations. Operations: Multiplication and Division. Number Sense and Numeration

Planning For Success Mathematics: Numeration Inquiry Investigations Operations: Multiplication and Division Number Sense and Numeration OVERALL EXPECTATIONS By the end of Grade 4, students will: solve

### Mathematics. Steps to Success. and. Top Tips. Year 5

Pownall Green Primary School Mathematics and Year 5 1 Contents Page 1. Multiplication and Division 3 2. Positive and Negative Numbers 4 3. Decimal Notation 4. Reading Decimals 5 5. Fractions Linked to

### Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

### Mathematics standards

Mathematics standards Grade 6 Summary of students performance by the end of Grade 6 Reasoning and problem solving Students represent and interpret routine and non-routine mathematical problems in a range

### Fractions as Numbers INTENSIVE INTERVENTION. National Center on. at American Institutes for Research

National Center on INTENSIVE INTERVENTION at American Institutes for Research Fractions as Numbers 000 Thomas Jefferson Street, NW Washington, DC 0007 E-mail: NCII@air.org While permission to reprint this

### 1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

### CALCULATIONS. Understand the operation of addition and the related vocabulary, and recognise that addition can be done in any order

CALCULATIONS Pupils should be taught to: Understand the operation of addition and the related vocabulary, and recognise that addition can be done in any order As outcomes, Year 1 pupils should, for example:

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

### A Prime Investigation with 7, 11, and 13

. Objective To investigate the divisibility of 7, 11, and 13, and discover the divisibility characteristics of certain six-digit numbers A c t i v i t y 3 Materials TI-73 calculator A Prime Investigation

### Multiplication and Division with Rational Numbers

Multiplication and Division with Rational Numbers Kitty Hawk, North Carolina, is famous for being the place where the first airplane flight took place. The brothers who flew these first flights grew up

### Paper 1. Calculator not allowed. Mathematics test. First name. Last name. School. Remember KEY STAGE 3 TIER 4 6

Ma KEY STAGE 3 Mathematics test TIER 4 6 Paper 1 Calculator not allowed First name Last name School 2007 Remember The test is 1 hour long. You must not use a calculator for any question in this test. You

T276 Mathematics Success Grade 6 [OBJECTIVE] The student will add and subtract with decimals to the thousandths place in mathematical and real-world situations. [PREREQUISITE SKILLS] addition and subtraction

### Review: Comparing Fractions Objectives To review the use of equivalent fractions

Review: Comparing Fractions Objectives To review the use of equivalent fractions in comparisons. www.everydaymathonline.com epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters

### MMLA Student Test/MathAssessments.MSCenters.Org. MMLA Mathematics Assessment Items

Page 1 of 42 MMLA Mathematics Assessment Items Name: Date: Multiple Choice Questions Select the one best answer for each question. 1. Which of the following sets of numbers are all of the factors of 24?

### Test 4 Sample Problem Solutions, 27.58 = 27 47 100, 7 5, 1 6. 5 = 14 10 = 1.4. Moving the decimal two spots to the left gives

Test 4 Sample Problem Solutions Convert from a decimal to a fraction: 0.023, 27.58, 0.777... For the first two we have 0.023 = 23 58, 27.58 = 27 1000 100. For the last, if we set x = 0.777..., then 10x

### Algebra Word Problems

WORKPLACE LINK: Nancy works at a clothing store. A customer wants to know the original price of a pair of slacks that are now on sale for 40% off. The sale price is \$6.50. Nancy knows that 40% of the original

### Securing number facts, calculating, identifying relationships

1 of 19 The National Strategies Primary Year 4 Block E: Three 3-week units Securing number facts, calculating, identifying relationships Tables 10 10; multiples Written methods: TU U; TU U; rounding remainders

### Lesson 1: Fractions, Decimals and Percents

Lesson 1: Fractions, Decimals and Percents Selected Content Standards Benchmarks Addressed: N-2-H Demonstrating that a number can be expressed in many forms, and selecting an appropriate form for a given