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Name• Estimating Perimeter, Area, and Volume<strong>Reteach</strong>ingLesson <strong>111</strong><strong>111</strong>• We can use grids to estimate the areas and perimeters of shapes that are notregular polygons, or when we don’t know the dimensions of the shape.ABCDEFG• To estimate the area of the triangle, count the whole squares (F) and the mostly fullsquares (E, B, G).• Do not count squares that are mostly empty (D, A, C).• To estimate the perimeter, add the estimated lengths of the sides. The base is 4units. We can see that one side is a little more than 3 units. The third side is a littlemore than 2 units. So, we estimate the perimeter to be between 9 units and 10units.• One way to estimate the volume of a container is to fill the container with unitcubes and count the cubes.Practice:Estimate the area of each figure on these grids. Each small square represents onesquare inch. Remember to write the units.1. 2. 3.Estimate the perimeter of each figure on these grids.4. 5.<strong>12</strong>2 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
Name• Reducing Fractions<strong>Reteach</strong>ingLesson 1<strong>12</strong>1<strong>12</strong>• Use mental math to divide the numerator (top number) and the denominator(bottom number) by the same number.Example: Reduce __ 8 (Hint: Divide 8 and <strong>12</strong> by 4)<strong>12</strong>___ 8<strong>12</strong> = ________ 08 ÷ 4<strong>12</strong> ÷ 4 = __ 2 3Practice:Write the reduced form of each fraction. Some may already be reduced.1. 2 __6 = 2. 3 __6 = 3. 4 __8 =Hint: ÷ 2 Hint: ÷ 3 Hint: ÷ 44. __ 6 9 = 5. ___ 2<strong>12</strong> = 6. ___ 4<strong>12</strong> =7.4 ___10= 8. ___ 1030 = 9. ___ 710 =10. ___ 610 = 11. __ 6 8 = <strong>12</strong>. ___ 6<strong>12</strong> =Saxon Math Intermediate 4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. <strong>12</strong>3
Name<strong>Reteach</strong>ing• Multiplying a Three-Digit Number by a Two-Digit NumberLesson 113113• Use mental math to carry.• Indent using 0 as a placeholder.Practice:Multiply. Remember to write the dollar sign and decimal point in money problems.1. 256× 32+ 02. 435× 18+ 03. $1 .6 7× 23+ 04. 511× 30+ 05. $8 .0 4× 29+ 06. 543× 21+ 07. $4 .9 2× 36+ 08. 714× 35+ 0<strong>12</strong>4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
Name• Simplifying Fraction Answers<strong>Reteach</strong>ingLesson 114114• Change an improper fraction answer to a mixed number.• Always reduce fraction answers if possible.• Sometimes it helps to “break apart” a mixed number answer and change theimproper fraction to another mixed number.Example:3 __ 4 5+ 4 __ 3 57 __ 7 5 = 7 + 1 __ 2 5 = 8 __ 2 5Practice:Simplify the answer to each sum or difference. (Change improper fractions to mixednumbers, and reduce if possible.)1. 3 __5 + 3 __5 = 2. 5 __8 – 1 __8 =3. 2 3 __7+ 1 6 __74. 6 1 __4+ 3 3 __4= =5. 5 5 __9– 3 2 __96. 4 ___ 610+ 7 ___ 810= =Saxon Math Intermediate 4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. <strong>12</strong>5
Name• Renaming Fractions<strong>Reteach</strong>ingLesson 115115• Rename fractions using the loop method.Multiply the numbers in the loop and divide by the number outside the loop.Example:3__4 = ___ ?<strong>12</strong><strong>12</strong> × 3 = 36 3__4 = ___ 936 ÷ 4 = 9 <strong>12</strong>Practice:Complete each equivalent fraction.1. 1 __5 = ___152.2__3 = ___153. 4 __7 = ___144.1__2 = ___145.5__8 = ___246.___ 5<strong>12</strong> = ___247. 4 __5 = ___308. __ 5 6 = ___30<strong>12</strong>6 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
Name• Common Denominators<strong>Reteach</strong>ingLesson 116116• To rename fractions so that they have a common denominator:1. Look down the multiplication table column for each denominator.2. Find the smallest number both columns share.3. Rename. (Use the loop method.)Example:3__4 and __ 4 51. Go down the 4s column and the 5s column.2. Find that they both share a 20.3. Rename.3__4 = ___ ?2020 × 3 = 60 ___ 1560 ÷ 4 = 15 204__5 = ___ ?2020 × 4 = 80 ___ 1680 ÷ 5 = 16 20Practice:1. Rename __ 1 3 and __ 1 so that they have a common denominator of 21.71__3 = ___211__7 = ___2<strong>12</strong>. Rename __ 3 5 and __ 5 so that they have a common denominator of 30.63__5 = ___305__6 = ___30Rename each pair of fractions using their least common denominator.3. __ 2 3 = ___<strong>12</strong>4. 1 __4 = ___3__4 = ___<strong>12</strong>3__8 = ___Saxon Math Intermediate 4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. <strong>12</strong>7
Name<strong>Reteach</strong>ing• Rounding Whole Numbers Through Hundred MillionsLesson 117117• Review the Place Value chart.Place Valuehundred millionsten millionsmillionshundred thousandsten thousandsthousandshundredstensonesdecimal point, , .100,000,00010,000,000tenthshundredthsthousandths1,000,000100,00010,000100010010<strong>111</strong>0110011000• To round numbers through hundred millions:1. Underline the place value that you will be rounding to.2. Ask: Is the number to the right 5 or more? (5, 6, 7, 8, 9)Yes Add 1 to the underlined number.No The underlined number stays the same.4. Replace the numbers after the underlined number with zero.Examples:438,270 440,000 <strong>12</strong>,876,250 13,000,000427,063 400,000 427,063,000 400,000,000Practice:Round each number to the nearest ten thousand.1. 17,652 , 2. 31,984 ,Round each number to the nearest hundred thousand.3. 747,985 , 4. 658,103 ,Round 4,586,7<strong>12</strong> to the nearest million.5. , ,Round 46,029,984 to the nearest ten million.6. , ,Round 792,463,051 to the nearest hundred million.7. , ,<strong>12</strong>8 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
Name• Dividing by Two-Digit Numbers<strong>Reteach</strong>ingLesson 118118• Use zero as a placeholder.• When you multiply, if the answer is greater than the dividend, try a smaller number.• If the ones digit in the divisor is 4 or less, round it down before guessing.____ 0 044 ) 163 (Think:____ 0 040 ) 160 )• If the ones digit in the divisor is 5 or more, round it up before guessing.________46 ) 163 (Think: 50 ) 160 )• Then use the four division steps to solve (divide, multiply, subtract, bring down).Practice:Divide using long division.1.____ 0 022 ) <strong>12</strong>9 2.___ 017 ) 833.____ 0 057 ) 243 4.____ 0 072 ) 2845.____ 0 051 ) 299 6.___ 014 ) 637.____ 0 025 ) <strong>12</strong>1 8.____ 0 084 ) 642Saxon Math Intermediate 4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. <strong>12</strong>9
Name<strong>Reteach</strong>ingLesson 119• Adding and Subtracting Fractions with Different Denominators119• To add or subtract fractions that have different denominators, first rename thefractions so that they have common denominators.1. Find a common denominator.2. Rename. (Use the loop method.)3. Add or subtract the renamed fractions.Example:2__3 = __ 0 9+ __ 4 9 = __ 0 92__3 = __ 6 9+ __ 4 9 = __ 4 9___ 109 = 1 __ 1 9Practice:Find each sum or difference.1.1__4 = ____82.3__4 = ___+__ 3 8 = ____8– 1 __2 = ___3.2__5 = ___4.5__8 = ___+___ 310 = ___+ 1 __4 = ___5.4__9 = ___6.6__9 = ___+ 1 __3 = ___– 2 __3 = ___7.9 ___10 = ___8.7__8 = ___– 2 __5 = ___– 1 __4 = ___130 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
Name• Adding and Subtracting Mixed Numbers with DifferentDenominators<strong>Reteach</strong>ingLesson <strong>12</strong>0<strong>12</strong>0• To add or subtract mixed numbers with different denominators:1. Copy the problem vertically.2. Rename the fractions so that they have common denominators.3. Add or subtract the fraction side.4. Add or subtract the whole numbers.5. Reduce the fraction side.Example:Practice:3 __ 1 7 = 3 ___ 214+ 2 ___ 614 = 2 ___ 6145 ___ 814 = 5 __ 4 7Find each sum or difference. Reduce when possible.1. 2 __ 1 3 + 1 ___ <strong>12</strong>152 1 __3 = 2 ___+ 1 ___ <strong>12</strong>15 = 1 ___ <strong>12</strong>152. 5 __ 3 7 + 2 ___ 5145 3 __7 = ________+ 000000 = ____3. 4 __ 2 3 + 3 __ 6 94 2 __3 = ____4. 6 __ 1 8 + 3 __ 3 46 1 __8 = ________+ 000000 = ________+ 000000 = ____5. 3 __ 5 8 – 2 __ 1 43 __ 5 8 = 3 __ 5 86. 2 ___ 610 – 2 __ 2 52 ___ 610 = ____– 2 __ 1 4 = 2 ___ 0000____– 000000 = ____7. 9 ___ 11<strong>12</strong> – 7 __ 5 69 ___ 11<strong>12</strong> = ____8. 5 ___ 914 – 4 __ 3 75 ___ 914 = ________– 000000 = ____– 000 ____000 = ____Saxon Math Intermediate 4 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. 131
Name• Solving Balanced Equations<strong>Reteach</strong>ingInvestigation <strong>12</strong>• An equation states two quantities are equal.• A model for an equation is a balance scale.• All equations have an equal sign.• To balance and solve the equation we must follow the steps below:– Read the problem.– Identify type of problem (combining, separating).– Get the block with the letter by itself on one side. Remember what you do to oneside, you have to do the other.– Find the number that solves the equation.– Check your answer.N 3 43 3Inv. <strong>12</strong>Practice:1.N5552.33N6Equation:Equation:Solution: N = Solution: N =3.N53334.4448 NEquation:Equation:Solution: N = Solution: N =5.N886.N5 55Equation:Equation:Solution: N = Solution: N =132 © Harcourt Achieve Inc. and Stephen Hake. All rights reserved. Saxon Math Intermediate 4
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