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§7.1 Decimals 3/30/17 Today We’ll Discuss How do we read and write decimals? How do we convert decimals and fractions? How do we recognize when a decimal terminates? How do we order terminating decimals? What are Decimals? Definitions: A decimal represents a fraction whose denominator is a power of 10. Instead of writing the fraction, digits are placed after a decimal point to represent the fraction. Fractional part For example: Whole-number part What are Decimals? Just like before, each digit represents a different place value. However, now each digit after the decimal represents how many of each “10-power fraction” we have. The book differentiates between each form using the terms fraction form and decimal form Notation: When starting out, stress writing decimals that are less than one using a leading zero. 0.25 instead of .25. Examples Base-Ten Block Model for DECIMALS The main point of the base-ten blocks is that each piece (cube, rod, flat,...) is 10 times the size of the last. Since each place value in a decimal is still 10 times the next, we can use base-ten blocks, but in a different way. New Values Example Use base-ten blocks to model a) 2.34 Example Use base-ten blocks to model a) 2.34 b) 1.59 Example Use base-ten blocks to model a) 2.34 b) 1.59 c) .15 Terminating vs. NonTerminating Decimals There are three main kinds of decimals: 1) Terminating 2) Non-Terminating, Repeating 3) Non Terminating, Non-Repeating Terminating Decimals Fact: A fraction in simplest form can be written as a terminating decimal if and only if the prime factorization of the denominator contains no other primes other than 2 or 5. Terminating Decimals There are 2 ways to write these fractions as terminating decimals. A) Rewrite the fraction so that its denominator is a power of 10 B) Divide the numerator by the denominator (mostly saved for §7.3) Example Determine whether each fraction can be written as a terminating decimal. If so, write the terminating decimal. Ordering Terminating Decimals As with many of the numbers we order, we use a number line to order decimals. 1) Draw a number line including all decimals to be ordered. 2) Graph each decimal on the number line. 3) Read the decimals from left to right to order from least to greatest. Example Order the following from least to greatest. a) 0.34, 0.4, 0.21, and 0.58 b) 0.3, 0.209, 0.25, and 0.2003 Example You are marking distances on a map from your home to several locations. Use a number line to decide which location is closest to your home. Example The results of the men’s 100 meter dash final at the 2012 Summer Olympics are shown below. Who won the gold, silver, and bronze medals? §7.2 Adding and Subtracting Decimals 3/30/17 Today We’ll Discuss How do we add and subtract decimals using models and algorithms? How do we add and subtract decimals using mental math? How do we round decimals? Base-Ten Block Addition/Subtraction (Decimals) We’ve already discussed the reasoning and method for base-ten blocks as a model for usual addition and subtraction. We can take all of that and transfer those skills directly to our new decimal values for base-ten blocks. Vertical Algorithm for Adding and Subtracting Decimals To add or subtract decimals: 1) Write the numbers vertically, lining up the decimal points. 2) Place the missing “invisible zeros.” 3) Add or subtract with your favorite old algorithm keeping the decimal throughout. 4) Carry down the decimal point into your final answer. Example Use the vertical algorithm to find a) 2.405 + 13.01 + 7.2 b) 5.3 − 1.255 c) 3.042 − 1.0322 Example The first and second place winning times for the women’s downhill Alpine skiing at a recent Winter Olympics were 1 minute, 44.19 seconds and 104.75 seconds, respectively. How much faster is the first place skier? Example Find the balance after each transaction. Example - Answer Find the balance after each transaction. Mental Math with Decimals Recall that we can use compatible numbers to make addition or subtraction easier mentally. The same concept still applies for decimals. Examples Compensation Strategy with Adding and Subtracting Decimals Also recall that compensation is a technique we used to change numbers in an addition or subtraction problem to compatible numbers. The key was: Addition → Add/Subtract oppositely Subtraction → Add/Subtract the same Example Use compensation to add or subtract the following. a) 2.8 + 2.3 b) 8.1 − 2.9 Example You purchase 5 packs of batteries online. The prices are $9.97, $11.97, $5.97, $6.97, and $18.72. You pay using the $75 gift card. How much credit is left on the gift card? Rounding Decimals All previous rounding rules still hold. We just need to be careful to round to the correct value. Examples Examples Rewrite each statement by rounding the decimal to a number that is reasonable in the real-life context. a) A typical weight of an adult Percheron horse is 1873.4 pounds. b) Adult giraffes generally stand between 4.572 and 6.096 meters tall. Homework #17 §7.1 - Pages 252-255 #4, 6, 8, 12, 18, 24, 40 §7.2 - Pages 262-265 #7, 10, 16