Lesson 1 Multiply multidigit whole numbers and multiples

Lesson 1: Multiply multi-digit whole numbers and multiples of 10 using place value patterns and the distributive and associative properties.

$� � � � � Decimal Fraction = a proper fraction whose denominator is$

� � � � � Decimal Fraction = a proper fraction whose denominator is a power of 10 Multiplier = a quantity by which a given number—a multiplicand—is to be multiplied Parentheses = the symbols used to relate order of operations Decimal = a fraction whose denominator is a power of ten and whose numerator is expressed by figures placed to the right of a decimal point Digit = a numeral between 0 and 9 Divisor = the number by which another number is divided Dividend = the number to be divided Equation = a statement that the values of two mathematical expressions are equal Equivalence = a state of being equal or equivalent Equivalent measures = e. g. , 12 inches = 1 foot; 16 ounces = 1 pound

� � � � � Estimate = approximation of the value of a quantity or number Exponent = the number of times a number is to be used as a factor in a multiplication expression Multiple = a number that can be divided by another number without a remainder like 15, 20, or any multiple of 5 Pattern = a systematically consistent and recurring trait within a sequence Product = the result of a multiplication Quotient = the answer of dividing one quantity by another Remainder = the number left over when one integer is divided by another Renaming = making a larger unit Rounding = approximating the value of a given number Unit Form = place value counting, e. g. , 34 stated as 3 tens 4 ones

�Say the product. � 3 x 100 � 3 x 1, 000 � 5 x 1, 000 � 0. 005 x 1, 000 � 50 x 100 � 0. 05 x 100 � 30 x 1, 000 � 32 x 1, 000 � 0. 32 x 1, 000 � 52 x 100 � 5. 2 x 100 � 4 x 10 � 0. 45 x 1, 000 � 30. 45 x 1, 000 � 7 x 100 � 72 x 100 � 7. 002 x 100

• • • 4 4 4 7 2 tens = ____ ten thousands = ____ hundred thousands = ____ millions = ____ thousands = _____ • • 3 tens = ____ 53 tens = ____ 6 ten thousands = ____ 36 ten thousands= ____

• Show the answer in a place value chart. • 8 hundred thousands 36 ten thousands = ____ • 8 millions 24 ten thousands = ____ • 8 millions 17 hundred thousands = ____ • 1034 hundred thousands = ____

• Use vertical number lines to round 8, 735 to the nearest thousands, hundreds, and tens places. • Use vertical number lines to round 7, 458 to the nearest thousands, hundreds, and tens places.

The top surface of a desk has a length of 5. 6 feet. The length is 4 times its width. What is the width of the desk? Use a tape diagram to model your problem. Check your answer using a standard algorithm. Be sure to include a statement of solution. Desk Length 4 5. 6 feet ? Desk Width The width of the desk is 1. 4 feet.

• 4 x 30 • 4 x 3 tens = _______ • What is 12 tens in standard form? • 120

• 4 tens x 3 tens = ____. Solve with a partner. • How did you use the previous problem to help you solve 4 tens x 3 tens? • The only difference was the place value unit of the first factor, so it was 12 hundreds. • It’s the same as 4 threes times 10, which is 12 hundreds. • You multiply 4 x 3, which is 12. Then multiply ten by ten, so the new units are hundreds. Now we have 12 hundreds, or 1200. • We can think of this problem as (4 x 3) x

• 4 tens x 3 hundreds = ____. • How is this problem different than the last problem? • We are multiplying tens and hundreds, not ones and hundreds, or tens and tens. • 4 tens is the same as 4 times 10. • 4 x 10 = 40 • 3 hundreds is the same as 3 times what? • 100 • 4 x 10 3 x 100 • So, another way to write our problem would be (4 x 10) x (3 x 100). • (4 x 3) x (10 x 100) • Are these expressions equal? Why or why not? Turn and talk to your right shoulder

• (4 x 3) x (10 x 100) • Yes, these expressions are the same. We can multiply in any order, so they are the same. • What is 4 x 3? • 12 • What is 10 x 100? • 1000 • What is the product of 12 and 1, 000? • 12, 000

• 4 thousands x 3 tens = ____. • How is this problem different than the last problem? • We are multiplying tens and thousands. • 4 thousands is the same as 4 times 1000. • 4 x 1000 = 4000 • 3 tens is the same as 3 times 10 • 3 x 10 = 30 • 4 x 1000 3 x 10 • So, another way to write our problem would be (4 x 1000) x (3 x 10). • (4 x 3) x (1000 x 10) • Are these expressions equal? Why or why not? Turn and talk to your shoulder partner.

• (4 x 3) x (1000 x 10) • Yes, these expressions are the same. We can multiply in any order, so they are the same. • What is 4 x 3? • 12 • What is 1000 x 10? • 10000 • What is the product of 12 and 10, 000? • 120, 000

• 60 x 5 = ____. • (6 x 10) x 5 (6 x 5) x 10 • Are both of these equivalent to 60 x 5? Why or why not? Turn and talk to your shoulder partner. • When we change the order of the factors we are using the commutative (any-order) property. • When we group the factors differently we are using the associative property of multiplication. • Let’s solve (6 x 5) x 10 • 30 x 10 = 300 • For the next problem, use the properties and what you know about multiplying multiples of

• 60 x 50 = ____. • Work with your table to solve this problem in different ways. Explain your thinking using words. • I thought of 60 as 6 x 10 and 50 as 5 x 10. I rearranged the factors to see (6 x 5) and (10 x 10). I got 30 x 100 = 3, 000 • I first multiplied 6 x 5 and got 30. Then I multiplied by 10 to get 300, and then multiplied by 10 to get 3, 000. • In the last problem set the number of zeros in the product was exactly the same number of zeros in our factors. That doesn’t seem to be the case here. Why is that? • 6 x 5 is 30, then we have to multiply by 100.

• Think about what we have discussed and solve 60 x 500 and 60 x 5, 000 independently in your math journal. • 60 x 500 • (6 x 5) x (10 x 100) • 30 x 1, 000 • 30, 000 • 60 x 5, 000 • (6 x 5) x (10 x 1000) • 30 x 10, 000 • 300, 000

• In your math journal, find the product of 451 x 8 using any method. • How did you solve this problem? • Vertical algorithm • Distributive property (400 x 8 + 50 x 8 + 1 x 8) • What makes the distributive property useful here? Why does it help here, but we didn’t really use it in our other problems? Turn and talk to your table. • There are different digits in three place values instead of all zeros. If I break the number apart by unit, then I can use basic