Fraction Decimal Launch Slides
Day 1 Launch
Equivalent Fractions: Same Value, New Look
Visual Scaling
Core Principle
Multiply or divide the numerator and denominator by the exact same non-zero number.
\( \frac{2}{3} \)
Original
\(\times \frac{4}{4} =\)
\( \frac{8}{12} \)
Equivalent
Math Talk Challenge
Maya claims that \( \frac{3}{5} \) is equal to \( \frac{9}{15} \), but Leo says they cannot be equal because the numbers 9 and 15 are much bigger.
How would you prove Maya is correct using words or a quick sketch?
Focus: Identifying & generating equivalent fractions Packet Day 1
Day 2 Launch
Comparing Fractions: Benchmark & Denominators
Strategy Tool
Strategy 1 • The Benchmark \( \frac{1}{2} \)
Compare both fractions to \( \frac{1}{2} \). Is one fraction clearly less than half and the other greater than half?
\( \frac{3}{8} \)
< \( \frac{1}{2} \) (since \( \frac{4}{8} = \frac{1}{2} \))
<
\( \frac{5}{6} \)
> \( \frac{1}{2} \) (since \( \frac{3}{6} = \frac{1}{2} \))
Strategy 2 • Common Denominator
When fractions are on the same side of half, create equal parts by finding a common denominator.
Compare \( \frac{2}{3} \) and \( \frac{3}{5} \) using denominator 15:
\( \frac{2}{3} = \frac{10}{15} \) > \( \frac{3}{5} = \frac{9}{15} \)
Focus: Reasoning with benchmark fractions and common denominators Packet Day 2
Day 3 Launch
Unlike Denominators: Rename to Combine
Common Units
3-Step Action Plan
1
Find a common multiple for the denominators (the LCD).
2
Rename each fraction into an equivalent fraction.
3
Add or subtract numerators only. Keep the denominator!
Sample in Action Problem: \( \frac{1}{4} + \frac{2}{3} \)
Common multiple of 4 and 3 is 12
\( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \) & \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
\( \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \)
Always check: Can the sum or difference be simplified?
Focus: Adding and subtracting fractions with unlike denominators Packet Day 3
Day 4 Launch
The Bridge: Fractions to Decimals
Place Value
Base-Ten Denominators
Tenths \( \frac{7}{10} = 0.7 \)
Hundredths \( \frac{45}{100} = 0.45 \)
Thousandths \( \frac{125}{1000} = 0.125 \)
Scale Denominator to 10 or 100
If your denominator is 2, 4, 5, 20, or 25, scale it to a power of 10 first!
\( \frac{3}{4} \times \frac{25}{25} = \frac{75}{100} = 0.75 \)
\( \frac{4}{5} \times \frac{2}{2} = \frac{8}{10} = 0.8 \)
Say it out loud: "eight tenths" is both \( \frac{8}{10} \) and \( 0.8 \)!
Focus: Understanding place value and converting between representations Packet Day 4
Day 5 Launch
Decimal Operations: Precision & Alignment
Place Value Lineup
The Number One Rule
Always line up the decimal points vertically so like places add or subtract together!
Annex trailing zeros if one number has fewer decimal digits (e.g., write \( 6.2 \) as \( 6.20 \)).
Watch Regrouping with Zeros
Subtract: \( 12.00 - 4.65 \)
12.00
- 4.65
7.35
Check with estimation: \( 12 - 5 = 7 \). Our answer 7.35 is very reasonable!
Focus: Adding and subtracting decimals to the hundredths place Packet Day 5
Day 6 Launch
Multiplying Decimals: Count the Places
Mastery Finale
Multiplication Algorithm
1
Multiply as whole numbers (temporarily ignore decimal points).
2
Count the total decimal places in both factors.
3
Count that same number of places from the right in your product.
Worked Example: \( 2.4 \times 0.7 \)
\( 2.4 \) (1 decimal place) • \( 0.7 \) (1 decimal place)
Total decimal places: 2
Multiply wholes: \( 24 \times 7 = 168 \)
Place point: 1.68
Estimation check: \( 2 \times 0.7 \approx 1.4 \), so 1.68 makes total sense!
Focus: Multiplying decimals using standard algorithms and place value reasoning Packet Day 6
Fraction Decimal Practice Packet
Day 1 of 6
Equivalent Fractions Mastery
Name: Date:
Key Strategy: Multiply or divide both numerator and denominator by the same non-zero number to create equal value.
1 Scaling Equivalent Fractions
Standard 5.NF.A.1
Use multiplication to find two different fractions equivalent to \( \frac{3}{4} \). Show how you scaled each one.
Equivalent Fractions: and
2 Missing Value Puzzle
Standard 5.NF.A.1
Find the unknown values to make the statement true: \( \frac{2}{5} = \frac{\square}{20} = \frac{14}{\triangle} \)
\( \square = \) \( \triangle = \)
3 Simplifying to Lowest Terms
Standard 5.NF.A.1
Simplify the fraction \( \frac{18}{24} \) to its simplest form by dividing by the greatest common factor (GCF).
GCF: Simplest Form:
4 Real-World Application
Word Problem
Marcus ran \( \frac{4}{6} \) of a mile. Sophia ran \( \frac{8}{12} \) of a mile. Did they run the exact same distance? Explain your reasoning.
Answer:
Self-Check: I scaled both the numerator and denominator by the exact same value.
Page 1 of 6
Day 2 of 6
Comparing & Ordering Fractions
Name: Date:
Key Strategy: Use benchmark fractions like \( \frac{1}{2} \) first, or find a common denominator to compare equal-sized parts.
1 Benchmark Reasoning
Standard 5.NF.A.2
Compare \( \frac{3}{8} \) and \( \frac{5}{6} \) using the benchmark \( \frac{1}{2} \). Fill in \( < \), \( > \), or \( = \) and explain.
Statement: \( \frac{3}{8} \) \( \frac{5}{6} \)
2 Common Denominators
Standard 5.NF.A.1
Compare \( \frac{4}{7} \) and \( \frac{5}{9} \) by finding a common denominator for 7 and 9. Show the equivalent fractions.
Equivalent Forms: and
3 Ordering Three Fractions
Standard 5.NF.A.2
Order these fractions from least to greatest: \( \frac{2}{3} \), \( \frac{5}{12} \), \( \frac{3}{4} \).
Least: < < :Greatest
4 Recipe Problem
Word Problem
Chef Liam uses \( \frac{7}{10} \) cup of honey for muffins and \( \frac{3}{5} \) cup for granola bars. Which recipe uses more honey, and by how much?
More Honey: Difference:
Self-Check: I verified whether both fractions are less than, equal to, or greater than \( \frac{1}{2} \).
Page 2 of 6
Day 3 of 6
Adding & Subtracting Unlike Fractions
Name: Date:
Key Strategy: Rename fractions using the Least Common Denominator (LCD). Add or subtract numerators, keep the denominator.
1 Addition with Unlike Denominators
Standard 5.NF.A.1
Solve and write your answer in simplest form: \( \frac{1}{3} + \frac{2}{5} \)
Common Denominator: Sum:
2 Subtraction with Simplifying
Standard 5.NF.A.1
Solve and write your answer in simplest form: \( \frac{7}{8} - \frac{1}{4} \)
Difference in Simplest Form:
3 Mixed Number Challenge
Standard 5.NF.A.1
Find the sum: \( 2\frac{1}{2} + 1\frac{2}{3} \). Show how you add whole numbers and renamed fractions.
Sum as Mixed Number:
4 Fabric Project Problem
Word Problem
Camila had \( \frac{5}{6} \) yard of blue fabric. She used \( \frac{3}{4} \) yard for a craft. How much fabric does Camila have left?
Fabric Left:
Self-Check: I only added or subtracted numerators after denominators were identical!
Page 3 of 6
Day 4 of 6
Fractions to Decimals & Place Value
Name: Date:
Key Strategy: Convert denominators to powers of ten (10, 100) or read place value names aloud ("three tenths" = 0.3).
1 Base-Ten Conversions
Standard 5.NBT.A.3
Convert each: (a) \( \frac{7}{10} \) as a decimal • (b) \( \frac{36}{100} \) as a decimal • (c) \( 0.85 \) as a simplified fraction.
(a) (b) (c)
2 Scaling to 100
Standard 5.NBT.A.3
Convert \( \frac{3}{4} \) and \( \frac{9}{20} \) into decimals by creating equivalent fractions with denominator 100.
\( \frac{3}{4} = \) \( \frac{9}{20} = \)
3 Comparing Values
Standard 5.NBT.A.3.b
Compare each pair using \( < \), \( > \), or \( = \): Pair A: \( 0.65 \) and \( \frac{3}{5} \) • Pair B: \( \frac{7}{10} \) and \( 0.07 \)
Pair A: \( 0.65 \) \( \frac{3}{5} \) Pair B: \( \frac{7}{10} \) \( 0.07 \)
4 Metric Problem
Word Problem
A science experiment requires \( \frac{1}{2} \) liter of water. Elena measured \( 0.55 \) liter. Did she pour too much or too little, and by how much?
Outcome: Difference:
Self-Check: I matched place values: tenths with tenths, hundredths with hundredths!
Page 4 of 6
Day 5 of 6
Adding & Subtracting Decimals
Name: Date:
Key Strategy: Line up decimal points vertically. Annex placeholder zeros so both numbers have the same number of decimal digits.
1 Vertical Alignment Addition
Standard 5.NBT.B.7
Line up vertically and solve: \( 14.85 + 6.3 \) (Remember to annex a placeholder zero!)
Sum:
2 Regrouping Across Zeros
Standard 5.NBT.B.7
Line up vertically and solve: \( 20.00 - 8.47 \) Show all regrouping steps clearly.
Difference:
3 Missing Value Mystery
Standard 5.NBT.B.7
Find the unknown value \( n \): \( n + 4.75 = 11.2 \)
\( n = \)
4 Store Receipt Problem
Word Problem
Aiden bought a notebook for $3.49 and pens for $5.75. He paid with a $20 bill. How much change did Aiden receive?
Total Cost: Change Back:
Self-Check: Did you bring your decimal point straight down into your answer?
Page 5 of 6
Day 6 of 6
Multiplying Decimals & Capstone
Name: Date:
Key Strategy: Multiply whole numbers first, then count the total decimal places in both factors to place the point.
1 Tenths by Tenths Product
Standard 5.NBT.B.7
Calculate the product: \( 0.6 \times 0.7 \) Explain how you determined the position of the decimal point.
Total Decimal Places: Product:
2 Standard Algorithm
Standard 5.NBT.B.7
Solve using the standard algorithm: \( 3.4 \times 2.5 \) Show all partial products clearly.
Product:
3 Reasoning & Estimation
Standard 5.NBT.B.7
Without multiplying fully, estimate the answer to \( 4.9 \times 6.2 \). Then find the exact product and compare.
Estimated: Exact:
4 Garden Area Project
Word Problem
A rectangular school garden plot measures \( 4.5 \) meters long and \( 2.8 \) meters wide. What is the total area of the garden in square meters?
Area:
6-Day Fraction & Decimal Quest Complete!
Page 6 of 6
Fraction Decimal Solutions Guide
Day 1 Key
Equivalent Fractions • Worked Solutions
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NF.A.1 Goal: Scaling by a fraction equal to 1 (\( \frac{n}{n} \))
Problem 1 • Scaling Equivalent Fractions Key: \( \frac{6}{8} \) and \( \frac{9}{12} \) (or any valid equivalent)
Worked Steps: Multiply \( \frac{3}{4} \times \frac{2}{2} = \frac{6}{8} \); Multiply \( \frac{3}{4} \times \frac{3}{3} = \frac{9}{12} \) (or \( \frac{12}{16} \), \( \frac{15}{20} \), etc.).
Diagnostic Note: Watch for students who add the same number to numerator and denominator (e.g., \( \frac{3+2}{4+2} = \frac{5}{6} \)). Remind them that equivalence requires multiplying by a form of 1.
Problem 2 • Missing Value Puzzle Key: \( \square = 8 \), \( \triangle = 35 \)
Worked Steps: Since \( 5 \times 4 = 20 \), multiply numerator by 4: \( 2 \times 4 = 8 \). For \( \frac{14}{\triangle} \), since \( 2 \times 7 = 14 \), multiply denominator by 7: \( 5 \times 7 = 35 \).
Diagnostic Note: If students struggle with the second ratio, advise them to compare directly back to the original base ratio \( \frac{2}{5} \) rather than the intermediate fraction.
Problem 3 • Simplifying to Lowest Terms Key: GCF = 6 • Simplest Form = \( \frac{3}{4} \)
Worked Steps: Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Greatest common factor is 6. Divide: \( 18 \div 6 = 3 \) and \( 24 \div 6 = 4 \). Result is \( \frac{3}{4} \).
Diagnostic Note: Students who divide by 2 get \( \frac{9}{12} \). Acknowledge it is equivalent, but point out 9 and 12 can still be divided by 3 to reach simplest terms.
Problem 4 • Real-World Word Problem Key: Yes, both ran \( \frac{2}{3} \) mile
Worked Steps: Simplify both or scale Marcus's: \( \frac{4}{6} \times \frac{2}{2} = \frac{8}{12} \). Alternatively, both reduce to \( \frac{2}{3} \). Therefore, Marcus and Sophia ran the exact same distance.
Diagnostic Note: Look for complete verbal or mathematical justifications. Simply stating "yes" without proof should receive partial credit.
Scoring: 4 points total (1 pt per complete justification and calculation) Guide Page 1 of 6
Day 2 Key
Comparing & Ordering Fractions • Worked Solutions
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NF.A.2 Goal: Benchmark reasoning (\( 1/2 \)) and common denominators
Problem 1 • Benchmark Reasoning Key: \( \frac{3}{8} < \frac{5}{6} \)
Worked Steps: Half of 8 is 4, so \( \frac{3}{8} < \frac{4}{8} = \frac{1}{2} \). Half of 6 is 3, so \( \frac{5}{6} > \frac{3}{6} = \frac{1}{2} \). Since \( \frac{3}{8} < \frac{1}{2} \) and \( \frac{5}{6} > \frac{1}{2} \), \( \frac{3}{8} < \frac{5}{6} \).
Diagnostic Note: Emphasize that finding a common denominator (24) is also mathematically valid, but benchmark reasoning is much faster mental math.
Problem 2 • Common Denominator Showdown Key: \( \frac{36}{63} > \frac{35}{63} \implies \frac{4}{7} > \frac{5}{9} \)
Worked Steps: Common denominator of 7 and 9 is 63. \( \frac{4 \times 9}{7 \times 9} = \frac{36}{63} \). \( \frac{5 \times 7}{9 \times 7} = \frac{35}{63} \). Comparing numerators: 36 > 35, so \( \frac{4}{7} > \frac{5}{9} \).
Diagnostic Note: Students often mistakenly guess \( \frac{5}{9} \) is bigger because both 5 and 9 are larger numerals than 4 and 7. The common denominator proves the contrary!
Problem 3 • Ordering Three Fractions Key: \( \frac{5}{12} < \frac{2}{3} < \frac{3}{4} \)
Worked Steps: Use common denominator 12: \( \frac{5}{12} = \frac{5}{12} \); \( \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \); \( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \). Ordering from least: \( \frac{5}{12} < \frac{8}{12} < \frac{9}{12} \).
Diagnostic Note: Make sure students write the original fractions in their final answer line, not just the renamed versions.
Problem 4 • Recipe Problem Key: Muffins; by \( \frac{1}{10} \) cup
Worked Steps: Rename \( \frac{3}{5} \) with denominator 10: \( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \) cup. Compare: \( \frac{7}{10} > \frac{6}{10} \) (Muffins use more). Difference: \( \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \) cup.
Diagnostic Note: Verify that students answered both questions (which recipe AND the difference amount).
Scoring: 4 points total (1 pt per correct comparison with visible equivalent model) Guide Page 2 of 6
Day 3 Key
Adding & Subtracting Unlike Fractions • Key
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NF.A.1 Goal: Finding LCD and maintaining equal parts during operations
Problem 1 • Addition with Unlike Denominators Key: LCD = 15 • Sum = \( \frac{11}{15} \)
Worked Steps: LCD of 3 and 5 is 15. \( \frac{1 \times 5}{3 \times 5} = \frac{5}{15} \). \( \frac{2 \times 3}{5 \times 3} = \frac{6}{15} \). Add numerators: \( \frac{5 + 6}{15} = \frac{11}{15} \). 11 is prime; cannot be simplified.
Diagnostic Note: Watch out for the common error \( \frac{1+2}{3+5} = \frac{3}{8} \). Emphasize that denominators represent partition size, not separate objects to add.
Problem 2 • Subtraction with Simplifying Key: \( \frac{5}{8} \)
Worked Steps: LCD of 8 and 4 is 8. Rename \( \frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8} \). Subtract: \( \frac{7}{8} - \frac{2}{8} = \frac{5}{8} \). Simplest form is \( \frac{5}{8} \).
Diagnostic Note: Highlight that one denominator (8) was already a multiple of the other (4), so only one fraction needed to be renamed.
Problem 3 • Mixed Number Challenge Key: \( 4\frac{1}{6} \) (or \( \frac{25}{6} \))
Worked Steps: Add whole numbers: \( 2 + 1 = 3 \). Add fractions: LCD of 2 and 3 is 6. \( \frac{1}{2} = \frac{3}{6} \); \( \frac{2}{3} = \frac{4}{6} \). \( \frac{3}{6} + \frac{4}{6} = \frac{7}{6} = 1\frac{1}{6} \). Combine: \( 3 + 1\frac{1}{6} = 4\frac{1}{6} \).
Diagnostic Note: Students often leave their answer as \( 3\frac{7}{6} \). Remind them that improper fraction parts must be converted and added to the whole number.
Problem 4 • Fabric Project Problem Key: \( \frac{1}{12} \) yard
Worked Steps: LCD of 6 and 4 is 12. Rename: \( \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \) and \( \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \). Subtract: \( \frac{10}{12} - \frac{9}{12} = \frac{1}{12} \) yard remaining.
Diagnostic Note: If students used 24 as the common denominator: \( \frac{20}{24} - \frac{18}{24} = \frac{2}{24} \), verify they simplified \( \frac{2}{24} \) to \( \frac{1}{12} \).
Scoring: 4 points total (Full credit requires common denominator work + correct final units) Guide Page 3 of 6
Day 4 Key
Fractions to Decimals & Place Value • Key
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NBT.A.3 Goal: Base-ten place values and converting representations
Problem 1 • Base-Ten Conversions Key: (a) 0.7 • (b) 0.36 • (c) \( \frac{17}{20} \)
Worked Steps: (a) "Seven tenths" = 0.7. (b) "Thirty-six hundredths" = 0.36. (c) \( 0.85 = \frac{85}{100} \). Divide numerator and denominator by 5: \( 85 \div 5 = 17 \), \( 100 \div 5 = 20 \implies \frac{17}{20} \).
Diagnostic Note: For (c), check if students left \( \frac{85}{100} \). Remind them that ending in 5 and 0 indicates divisibility by 5.
Problem 2 • Scaling to 100 Key: \( \frac{3}{4} = 0.75 \) • \( \frac{9}{20} = 0.45 \)
Worked Steps: \( \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75 \). For \( \frac{9}{20} \): multiply by \( \frac{5}{5} \implies \frac{9 \times 5}{20 \times 5} = \frac{45}{100} = 0.45 \).
Diagnostic Note: Useful money anchor: Remind students that \( \frac{3}{4} \) is 3 quarters out of 4, which equals 75 cents ($0.75).
Problem 3 • Comparing Values Key: Pair A: \( 0.65 > \frac{3}{5} \) • Pair B: \( \frac{7}{10} > 0.07 \)
Worked Steps: Pair A: Convert \( \frac{3}{5} = \frac{6}{10} = 0.60 \). Since \( 0.65 > 0.60 \), \( 0.65 > \frac{3}{5} \). Pair B: \( \frac{7}{10} = 0.70 \) (7 tenths). 0.07 is 7 hundredths. \( 0.70 > 0.07 \).
Diagnostic Note: In Pair B, students frequently think \( 0.7 = 0.07 \) because both contain 7. Stress place value: tenths are 10 times greater than hundredths.
Problem 4 • Metric Problem Key: Too much; by 0.05 liter
Worked Steps: Convert \( \frac{1}{2} = 0.5 = 0.50 \) liter. Elena poured \( 0.55 \) liter. \( 0.55 > 0.50 \), so she poured too much water. Difference: \( 0.55 - 0.50 = 0.05 \) liter (5 hundredths of a liter).
Diagnostic Note: If students wrote 0.5 instead of 0.05, they subtracted without aligning place value columns.
Scoring: 4 points total (Check proper decimal place labeling) Guide Page 4 of 6
Day 5 Key
Adding & Subtracting Decimals • Key
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NBT.B.7 Goal: Vertical decimal alignment and regrouping across zeros
Problem 1 • Vertical Alignment Addition Key: 21.15
Worked Steps: Annex zero to 6.3 \( \rightarrow 6.30 \). Line up points: \( 14.85 + 06.30 \). Hundredths: \( 5 + 0 = 5 \). Tenths: \( 8 + 3 = 11 \) (write 1, regroup 1). Ones: \( 1 + 4 + 6 = 11 \) (write 1, regroup 1). Tens: \( 1 + 1 = 2 \). Sum = 21.15.
Diagnostic Note: Students who don't line up points often add \( 14.85 + 6.3 \) right-aligned, resulting in \( 15.48 \). Have them draw vertical lines through decimal points.
Problem 2 • Regrouping Across Zeros Key: 11.53
Worked Steps: Setup: \( 20.00 - 8.47 \). Regroup from tens: 2 tens become 1 ten, 9 ones, 9 tenths, 10 hundredths. Hundredths: \( 10 - 7 = 3 \). Tenths: \( 9 - 4 = 5 \). Ones: \( 9 - 8 = 1 \). Tens: \( 1 - 0 = 1 \). Difference = 11.53.
Diagnostic Note: Common error: writing 7 and 4 down because of subtracting from 0 (e.g., getting 12.47). Model using place value disks or base ten blocks.
Problem 3 • Missing Value Mystery Key: \( n = 6.45 \)
Worked Steps: Isolate \( n \) using inverse operation: \( n = 11.2 - 4.75 \). Annex zero: \( 11.20 - 4.75 \). Hundredths: \( 10 - 5 = 5 \). Tenths: \( 11 - 7 = 4 \). Ones: \( 10 - 4 = 6 \). \( n = 6.45 \). Check: \( 6.45 + 4.75 = 11.20 \).
Diagnostic Note: Students who add instead of subtract get 15.95. Teach students to check whether substituting their value makes the equation true.
Problem 4 • Store Receipt Problem Key: Cost = $9.24 • Change = $10.76
Worked Steps: Step 1 (Total Cost): \( 3.49 + 5.75 = 9.24 \). Step 2 (Change): \( 20.00 - 9.24 = 10.76 \). Aiden received $10.76 in change.
Diagnostic Note: Two-step problem: check that students completed both steps and remembered to include the dollar sign ($) in both responses.
Scoring: 4 points total (1 pt per accurate calculation showing regrouping) Guide Page 5 of 6
Day 6 Key
Multiplying Decimals • Worked Solutions
Teacher Guide
Focus Standard: CCSS.MATH.CONTENT.5.NBT.B.7 Goal: Area model reasoning and standard multiplication algorithm
Problem 1 • Tenths by Tenths Product Key: 2 decimal places • Product = 0.42
Worked Steps: Multiply whole numbers: \( 6 \times 7 = 42 \). Count decimal places: 0.6 has 1 place (tenths), 0.7 has 1 place (tenths). Tenths \( \times \) tenths = hundredths (2 decimal places). Move point 2 places left: 0.42.
Diagnostic Note: Beware of answers like 4.2. Connect to fraction form: \( \frac{6}{10} \times \frac{7}{10} = \frac{42}{100} = 0.42 \).
Problem 2 • Standard Algorithm Key: 8.5 (or 8.50)
Worked Steps: Multiply \( 34 \times 25 \): Partial product 1: \( 34 \times 5 = 170 \). Partial product 2: \( 34 \times 20 = 680 \). Sum = \( 170 + 680 = 850 \). Count decimal places: 3.4 (1) + 2.5 (1) = 2 places. Place point: 8.50 = 8.5.
Diagnostic Note: Students often drop the trailing zero before counting decimal places (getting 0.85). Emphasize counting places on 850 first, then simplifying 8.50 to 8.5.
Problem 3 • Reasoning & Estimation Key: Estimate ≈ 30 • Exact = 30.38
Worked Steps: Round to nearest whole: \( 4.9 \approx 5 \) and \( 6.2 \approx 6 \). Estimated product: \( 5 \times 6 = 30 \). Exact calculation: \( 49 \times 62 = 3038 \). With 2 decimal places, exact product is 30.38. 30.38 is very close to 30.
Diagnostic Note: Estimation is an essential sanity check that prevents students from placing decimal points at 3.038 or 303.8.
Problem 4 • Garden Area Project Key: 12.6 square meters
Worked Steps: Area = length \( \times \) width = \( 4.5 \times 2.8 \). Multiply \( 45 \times 28 \): \( 45 \times 8 = 360 \); \( 45 \times 20 = 900 \). \( 360 + 900 = 1260 \). Place 2 decimal digits: 12.60 sq meters = 12.6 sq meters.
Diagnostic Note: Ensure students include units of area (square meters, \( \text{m}^2 \)) rather than linear meters.
Unit Mastery Goal: 80%+ on daily 4-question checks Guide Page 6 of 6
