The Lattice Method of Multiplication


 Stewart Perkins
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1 The Lattice Method of Multiplication Objective To review and provide practice with the lattice method for multiplication of whole numbers and decimals. epresentations etoolkit Algorithms Practice EM Facts Workshop Game Family Letters Assessment Management Common Core State Standards Curriculum Focal Points Interactive Teacher s Lesson Guide Teaching the Lesson Ongoing Learning & Practice Differentiation Options Key Concepts and Skills Apply multiplication facts. [Operations and Computation Goal ] Use the lattice method for multiplying whole numbers and decimals. [Operations and Computation Goal ] Use magnitude estimates to verify lattice method solutions. [Operations and Computation Goal ] Key Activities Students review and practice the lattice method for multiplying multidigit whole numbers. They practice the lattice method for multiplying decimals. Ongoing Assessment: Informing Instruction See page 9. Ongoing Assessment: Recognizing Student Achievement Use an Exit Slip (Math Masters, page ). [Operations and Computation Goal ] Key Vocabulary lattice method lattice Playing Factor Bingo Math Masters, p. per group: each of number cards 9 (from the Everything Math Deck, if available) counters Students practice applying multiplication facts and number properties, recognizing factors, and factoring numbers. Multiplying and Dividing by Powers of Math Journal, pp. A and B Students practice multiplying and dividing by powers of. Math Boxes 9 Math Journal, p. Students practice and maintain skills through Math Box problems. Study Link 9 Math Masters, p. 9 Students practice and maintain skills through Study Link activities. ENRICHMENT Exploring an Ancient Multiplication Method Math Masters, p. Students analyze an ancient multiplication method and use that method to solve a multiplication problem. EXTRA PRACTICE Minute Math Minute Math, p. Students practice multiplying decimals. EXTRA PRACTICE Multiplying Decimals Math Masters, p. Students practice multiplying decimals and writing about their strategies. Materials Math Journal, pp. and Student Reference Book, pp. and Study Link Math Masters, pp.,, and Class Data Pad Advance Preparation For Part, draw a by and a by lattice on the board. You might find it helpful to copy the lattice computation grids on Math Masters, page 7 on a transparency. Each student will need copies of Math Masters, page. Teacher s Reference Manual, Grades pp. Unit Estimation and Computation
2 Getting Started Mathematical Practices SMP, SMP, SMP, SMP Content Standards.NBT.,.NBT.7 Mental Math and Reflexes Write problems on the board or the Class Data Pad so students can visually recognize the patterns of decimal point placement in the products. Ask students to describe the patterns they notice in each set Math Message Study the problems in Column A on journal page. Then use lattice multiplication to solve the problems in Column B. Study Link FollowUp Allow students five minutes to work together to compare their answers and correct any errors. Teaching the Lesson Math Message FollowUp (Math Journal, p. ) WHOLECLASS DISCUSSION Algorithm Project The focus of this lesson is the lattice method for multiplying whole numbers and decimals. To teach U.S. traditional multiplication with whole numbers and with decimals, see Algorithm Project on page A and Algorithm Project on page A7. Ask volunteers to demonstrate on the board how they solved the problems. Note students explanations of their processes, including their vocabulary, for use in the following review. Reviewing the Lattice Method of Multiplication (Student Reference Book, pp. and ; Math Journal, p. ; Math Masters, p. ) WHOLECLASS The lattice method for multiplying has been used since Third Grade Everyday Mathematics. This method is very easy to use because it relies almost entirely on the recall of the basic multiplication facts. Use the student examples on the board from the Math Message FollowUp to emphasize the vocabulary of the lattice method. The box with squares and diagonals is called a lattice. Ask students how they know what size lattice to use and where to write the factors. The factors tell which size lattice to use. Write the second factor above the lattice and the first factor on the right side. Some students might think of the rectangular array model for multiplication: The number of digits in the first factor is the number of rows, and the number of digits in the second factor is the number of columns. Multiply the digits. Write the answers. Ask students how they know where to write these partial products. Write the answers in the square where the multiplied digits intersect on the lattice, digit per diagonal. Some students may note that each partial product is never more than digits: Tens go in the left diagonal and ones go in the right diagonal. 9 Lattice Practice Study the problems and solutions in Column A. Then use lattice multiplication to solve the problems in Column B. Column A Column B Example : =, Example : 7 9 =, s s,s,s s s,s,s 9 7 Math Journal, p =,9 s s,s,s =, s s,s,s 7 EMcuGMJ_U_99.indd // : AM Lesson 9 7
3 Whole Numbers Lattice Method The lattice method for multiplying has been used for hundreds of years. It is very easy to use if you know the basic multiplication facts. * =? The box with cells and diagonals is called a lattice. Write above the lattice. Write on the right side of the lattice. Multiply *. Then multiply *. Then multiply *. Write the answers as shown. Add the numbers along each diagonal, starting at the right. Read the answer. * =,9 * 7 =? Write 7 above the lattice. Write on the right side of the lattice. Multiply * 7. Then multiply *. Multiply * 7. Then multiply *. Write the answers as shown. Add the numbers along each diagonal, starting at the right. When the numbers along a diagonal add up to or more: record the ones digit in the sum. add the tens digit to the sum along the next diagonal above. Read the answer. * 7 =, 9 Add the numbers along each diagonal. Ask students how they know where to begin the addition. Begin with the diagonal in the bottom righthand corner. Explain that the extended diagonal lines outside of the lattice squares are the answer spaces. Ask students how the sum is written when the sum on a diagonal is or more. Write the ones digit in the answer space and the tens digit at the top of the next diagonal. Review these steps by demonstrating with the following problem. The Lattice Method for Multiplying Whole Numbers Example: 7 =?. Write 7 above the lattice. Write on the right side. Draw a lattice for each problem. Then multiply.. * 7. *. 77 * 9. *. 7 * Check your answers on pages and. Student Reference Book, p.. Multiply 7. Then multiply. Multiply 7. Then multiply. Write the answers as shown. Decimals and Percents Lattice Multiplication with Decimals Find. *. using lattice multiplication. Step : Make a magnitude estimate.. *. * 7 The product will be in the tens. (The symbol means is about equal to.) Step : Draw the lattice and write the factors, including the decimal points, at the top and right side. In the factor above the grid, the decimal point should be above a column line. In the factor on the right side of the grid, the decimal point should be to the right of a row line. Step : Find the products inside the lattice. Steps Step : Add along the diagonals, moving from right to left. Step : Locate the decimal point in the answer as follows. Slide the decimal point in the factor above the grid down along the column line. Slide the decimal point in the factor on the right side of the grid across the row line. When the decimal points meet, slide the decimal point down along the diagonal line. Write a decimal point at the end of the diagonal line. 7 Step : Compare the result with the estimate. The product, 7.7, is very close to the estimate of 7. 7 Find 7. *. using lattice multiplication. A good magnitude estimate is 7. *. 7 * 7. (The symbol means is about equal to.) The product, 77.7, is close to the estimate of Steps The lattice method of multiplication was used by Persian scholars as long ago as the year. It was often called the grating method.. Add the numbers along each diagonal. Begin with the diagonal in the bottom righthand corner. The sum of the numbers in the second diagonal is, so write above the in the third diagonal. The sum of the numbers in that diagonal is + + =.. Stress that the answer is shown, starting on the upper left side of the lattice and continuing below the lattice. 7 =, Pass out two latticecomputation grids (Math Masters, page ) to each student. Students work with partners to use the lattice method to solve several multiplication problems in which both factors are whole numbers. Remind students to make a magnitude estimate for the problem before beginning the lattice. Suggestions: 7 7; 9; ;. Draw a lattice for each problem and multiply... *... * 7.. *. Check your answers on page. Student Reference Book, p. Unit Estimation and Computation
4 Ongoing Assessment: Informing Instruction Watch for students who are still having difficulty with the automatic recall of multiplication facts. Provide them with copies of the lattice multiplication facts table (Math Masters, page ). Name 9 Lattice Multiplication Table Teaching Master 7 The Lattice Method for Multiplying Decimals An advantage of the lattice method is that products of decimals are as easy to calculate as products of whole numbers, and the grid automatically locates the placement of the decimal point in the answer. When writing the factors above and on the right side of the lattice, include the decimal points. In the factor above the grid, the decimal point should be above a column line. In the factor on the right side of the grid, the decimal point should be to the right of a row line. Locate the decimal point in the answer as follows: Slide the decimal point in the factor above the grid down. Slide the decimal point in the factor on the right side of the grid across. The decimal points will intersect on a diagonal line. Slide that decimal point down along the diagonal line. Place a decimal point at the end of the diagonal line. Model how to use the lattice to locate the decimal point in problems with factors such as. and in problems where one factor is a whole number such as.7. Have students work with partners to use the lattice method to solve several multiplication problems in which at least one factor is a decimal. Remind students to make a magnitude estimate for the problem before beginning the lattice. Suggestions: 7.7.9;. 9..;. 7.;...7. Multiplying Whole Numbers and Decimals by the Lattice Method (Math Journal, p. ) Students solve the problems independently and then check each other s work. Circulate and assist. Bring the class together to share magnitude estimates and solutions and to explain strategies and the reasoning used = Math Masters, p. 9 For each problem: Multiplication by the Lattice Method Make a magnitude estimate. Circle the appropriate box. Solve the problem using the lattice method. Show your work below., ,...., s s,s,s s s,s,s. s s,s,s s s,s,s s s s,s s s,s,s Math Journal, p. Lesson 9 9
5 9 Math Boxes. Give the value of the boldface digits in each numeral. a. 9. hundredth b.,9,7 million c.,.7 tenths d.,7.9 thousand. Write the prime factorization of 7.. Measure angle BOP to the nearest degree. B O BOP: P. Solve. a. n 9 n b. n 7 n 9 c.. p.9 p 7.. Solve. a., b., c., 7 d. 9, 7, e.,,,, 9. Cross out the shapes below that are not polygons. 9 Ongoing Assessment: Recognizing Student Achievement Exit Slip Use an Exit Slip (Math Masters, page ) to assess students ability to use a multiplication method of their choice. Have students write a response to the following: Explain how to use the multiplication method of your choice to solve 7 =? and. 7. =? Students are making adequate progress if their explanations support their multiplication strategy and appropriate placement of the decimal. [Operations and Computation Goal ] Ongoing Learning & Practice Playing Factor Bingo (Math Masters, p. ) SMALLGROUP Math Journal, p. Students practice applying multiplication facts and number properties, recognizing factors, and factoring numbers by playing Factor Bingo. See this lesson guide, page, for game directions. It can also be played as a wholeclass activity. Multiplying and Dividing by Powers of (Math Journal, pp. A and B) Students practice multiplying and dividing by powers of. Study Link Master Name Math Boxes 9 (Math Journal, p. ) STUDY LINK 9 Multiply with the Lattice Method For each problem: Make a magnitude estimate. Circle the appropriate box. Solve using the lattice method. Show your work in the grids.. 9 º 7 s s,s,s.. º.,.7.s s s s. º.7.s s s s. º., s,s,s,s Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson . The skills in Problems and preview Unit content. Study Link 9 (Math Masters, p. 9) Home Connection Students make estimates and then solve multiplication problems by the lattice method º 9...s s s s Practice. 7,,7 9, R $.7 $. $9. Math Masters, p. 9 Unit Estimation and Computation
6 Differentiation Options ENRICHMENT PARTNER Exploring an Ancient Multiplication Method (Math Masters, p. ) Min To apply students understanding of multiplication, have them explore and analyze how to use an ancient Egyptian method of multiplying. When students have finished, have them describe the pattern that is used in the Egyptian method. 9 Dividing by Powers of Use the strategy described in Extended Division Facts on page of the Student Reference Book to solve Problems. Example:, / =? Divide a whole number that ends in zero by a power of., / =, Count the number of zeros in the divisor (power of )., / =, Cross out the same number of zeros in the answer (quotient) as there are zeros in the divisor (power of ), starting in the ones place., / = The numbers not crossed out give the answer. / =. 7, / =.,,. 9, / = 9,., /, =., /, = 9., / = 7 Solve the following problems. Show your work. 7. Every year, Mrs. Connick asked her students to bring in bottle caps for charity. It took Mrs. Connick years to collect one million bottle caps. About how many bottle caps did each class collect every year?, bottle caps. The entire fifth grade collected about, buttons for their Math Fun Fair estimation jar. There are fifthgrade students. If each student collected the same number of buttons, about how many buttons did each student bring in? buttons NOTE This method is also used in Project : An Ancient Multiplication Algorithm. See page in this lesson guide. EXTRA PRACTICE SMALLGROUP Minute Math Min To offer students more experience with multiplying decimals, see Minute Math, page. 9. Elves love to bake cookies. Last year, they had a contest to see who could bake the most cookies in a week. Effie, the fastest elf, baked 7 cookies an hour. She baked for hours per day. About how many cookies did she bake in a week?,9 cookies Math Journal, p. A EMcuGMJ_U_99.indd A // 9: AM EXTRA PRACTICE Multiplying Decimals (Math Masters, p. ) Min To offer students more experience with decimal multiplication, have them record the following problems onto an Exit Slip, Math Masters, page. Ask students to make magnitude estimates for each product and then solve the problems. Students should use their estimates to ensure that their answers are reasonable. Have them write an explanation of their reasoning in solving Problem a. a... =. c... =.9 b.. =, d...9 =. 9 Multiplying and Dividing Decimals by Powers of Study the problems below. 7 º = º = 7 9 º = 7 9 º =. Explain a strategy for finding a product when you multiply a number by a power of greater than. Do the same for a power of less than. Sample answer: When multiplying by,, or,, you move the decimal point to the right; when multiplying by.,., or., you move the decimal point to the left. Study the problems below. = 7 = 7 9, = 79, =. Explain a strategy for finding the quotient when you divide a number by a power of greater than. Sample answer: When dividing by,, or,, you move the decimal point one place to the left for each in the power of.. Use your strategies to solve the problems below. a. 9.7 = 9.7 b..77, =,77 c.. _. = d. 99. _ 9.9 = e.. =. f. 9,, = 9. Math Journal, p. B EMcuGMJ_U_99.indd B // 9: AM Lesson 9
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