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PRE-ALGEBRA Pre-Algebra Coordinate System and Functions Ratios and Proportions Other Types of Numbers Coordinate System and Functions Instruction begins in third or fourth grade. First students learn how to plot points on the coordinate system: X on horizontal axis Y on the vertical axis Where is (7,6)? Coordinate System and Functions Next, students learn to complete a table with the function given: x 0 Function x + 2 0+2 y 2 1 1+2 3 2 3 4 2+2 4 Coordinate System and Functions After completing the table, students plot the points and draw the line for the function. Coordinate System and Functions After several lessons completing a table with the function provided, students are shown how to derive the function when given two pairs of points—Format 20.1, page 453. Coordinate System and Functions Finally, students can be taught to derive the function when given the points on the coordinate system with a line draw through them. Ratios and Proportions What is the preskill for ratios and proportions? How should problems be set up? Ratios and Proportions Example set up using equivalent fractions preskill to solve ratio problems: The store has 3 TVs for every 7 radios. If there are 28 radios in the store, how many TVs are there? TVs = TVs Radios Radios 3 TVs = 7 Radios TVs 28 Radios Ratios and Proportions 3TVs = TVs 7 Radios 28 Radios 3TVs (4 ) = TVs 7 Radios (4 ) 28 Radios Ratios and Proportions After reviewing the use of equivalent fractions, one may introduce problem solving with a ratio table. Ratios and Proportions A factory makes SUVs and cars. It makes 5 SUVs for every 3 cars. If the factory made 1600 vehicles last year, how many cars and how many SUVs did it make?: Classification Cars SUVs Vehicles Ratio 3 5 Quantity 1600 Ratios and Proportions After working with simple ratio tables, teachers may introduce tables for problems using fractions such as: Two-thirds of the people at Starbucks are drinking coffee. The rest are drinking tea. If 15 people are drinking tea, how many are drinking coffee? How many people are there in Starbucks? (p 449). Ratios and Proportions Students: 1) set up the ratio table and 2) complete the fraction family column: Coffee Fraction family 2/3 Tea 1/3 People 3/3 Ratios Quantity Ratios and Proportions 3) Students use the numerator of the fraction to complete the ratio column. Coffee Fraction family 2/3 Ratios Tea 1/3 1 People 3/3 3 2 Quantity Ratios and Proportions 4) Students fill in known quantities. Coffee Fraction family 2/3 Ratios Tea 1/3 1 People 3/3 3 Quantity 2 15 Ratios and Proportions 5) Students write the ratio equation: 2 Coffee 1 Tea = Coffee 15 Tea Ratios and Proportions 6) Students solve the ratio problem to answer questions. Coffee Fraction family 2/3 Tea People Ratios Quantity 2 30 1/3 1 15 3/3 3 Ratios and Proportions 7) Students use the number-family strategy to solve for unknowns. (See Format 20.2) Coffee Fraction family 2/3 Tea People Ratios Quantity 2 30 1/3 1 15 3/3 3 45 Ratios and Proportions Ratio and proportions can also be used to solve comparison problems like: Louise was paid 5/6 of what her boss was paid. If Louise is paid $1800 per month, how much more does her boss get paid, and what does her boss get paid? Ratios and Proportions Students set up a number family using fractions: Difference Louise Boss 1/6 + > 6/6 Ratios and Proportions Students can then use the ratio table and ratio equation to solve for the unknown quantities. Difference Louise 1 5 Boss 6 1800 Ratios and Proportions Ratio and Proportions can also be used to solve percentage problems such as: A store got 40% of its oranges from California and the rest from Florida. If the store had 170 total oranges, how many were from California and how many from Florida? Ratio and Proportions A store got 40% of its oranges from California and the rest from Florida. If the store had 170 total oranges, how many were from California and how many from Florida? First students complete the number family: California Florida All 40% + % >100% Ratios and Proportions A store got 40% of its oranges from California and the rest from Florida. If the store had 170 total oranges, how many were from California and how many from Florida? Students then put the information into a ratio table: California 40% Florida 60% All 100% 170 Ratios and Proportions Finally, students can use ratio tables to do comparison problems using percentages: A bike store sold 25% fewer women’s bicycles than men’s bicycles. If the store sold 175 fewer women’s bikes, how many men’s and women’s bikes did it sell? Ratios and Proportions A bike store sold 25% fewer women’s bicycles than men’s bicycles. If the store sold 175 fewer women’s bikes, how many men’s and women’s bikes did it sell? Again, students would start with the number family: Difference Women’s Men’s 25% + % > 100% Ratios and Proportions A bike store sold 25% fewer women’s bicycles than men’s bicycles. If the store sold 175 fewer women’s bikes, how many men’s and women’s bikes did it sell? Then the information from the number family would into the ratio table: Difference 25% Women’s 75 Men’s 100 175 Other Types of Numbers Primes and Factors Integers Exponents Other Types of Numbers Primes and Factors What are prime numbers? How do students “test” numbers to determine if they are prime? What examples should one use for this activity? Other Types of Numbers Primes and Factors What are the prime factors of a number? How can the prime factors of a number be determined? Other Types of Numbers Primes and Factors What are the prime factors of 30? What are prime factors used for? Other Types of Numbers Integers What are integers? How do the authors recommend introducing negative numbers? What is the rule? Other Types of Numbers Integers What is absolute value? How is this introduced to students? Once students understand the concept, students can solve problems with positive and negative integers, Format 20.3, p. 458. Other Types of Numbers Integers What rules does Format 20.3 teach? Other Types of Numbers Integers 1. 2. 3. 4. What rules does Format 20.3 teach? If the signs of the numbers are the same, you add. If the signs of the numbers are different, you subtract. When you subtract, you start with the number that is farther from zero and subtract the other number. The sign in the answer is always the sign of the number that is farther from zero. Other Types of Numbers Integers What rules do students need to know to multiply integers? Other Types of Numbers Integers What rules do students need to know to multiply integers? Plus x plus = plus; Minus x plus = minus; Minus x minus = plus; Plus x minus = minus Other Types of Numbers Exponents What is used initially to help students understand exponents? What is the base number? What is the exponent? 53 Other Types of Numbers Exponents How can multiplying numerals with exponents be shown? 4x4x4x4x4 43 x 42 = 45 Other Types of Numbers Exponents How can simplifying exponents be shown? 55 53 = 5x5x5x5x5 5x5x5