Master foundational fluency for Grade 6 mathematics. Explore 78 lessons across Units 1–9 with prerequisite skill progressions, 2-minute diagnostic quick checks, and interactive practice studios.
78 Lessons312 Prerequisite Skills9 Units (Units 1–9)624 Interactive Tasks
Showing 78 of 78 lessons
Unit 16 lessons
Unit 1: Math Is Discovery
Equal groups, friendly-ten estimates, fractions of a whole number, and decimal place value. Every argument, estimate, and pattern rule in this unit rests on multiplying with confidence and checking that the result is reasonable.
Lesson 1-1
Math is Mine
Doers of Math · Estimating Equal Groups · Decomposing Numbers
MP.3
Why this unlocks Lesson 1.1: This lesson builds on 4 skills from earlier math. Start with “Multiply a one-digit number by a multiple of 10”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a one-digit number by a multiple of 10Gr 4
2Round a two-digit number to the nearest ten to estimate a productGr 4
3Decide when equal groups call for multiplication, not additionGr 3
4Decompose a decimal into place-value parts that rebuild it exactlyGr 5
2-Minute Quick Check
20 × 4? Estimate 78 × 6 by rounding 78 to the nearest ten. Is 70 + 8.5 equal to 78.5?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 30 × 7.
✓ Answer: 210
Task 2: Round 47 to the nearest ten, then multiply by 5 to estimate 47 × 5.
✓ Answer: 250
Task 3: A van holds 9 students, and 6 vans go on a trip. Should you add 9 + 6 or multiply 9 × 6 to find how many students go? Explain.
✓ Answer: Multiply: there are 6 equal groups of 9, so 9 × 6 = 54 students. Adding 9 + 6 only counts one van and the number of vans.
Task 4: Write 64.3 as a tens part plus the rest, then add your parts to check.
✓ Answer: 60 + 4.3 = 64.3. The 6 is in the tens place, so it is worth 60.
Lesson 1-2
Math is Exploring and Thinking
Known and Unknown Quantities · Fractions of a Whole Number · Checking a Plan
5.NF.B.4
Why this unlocks Lesson 1.2: This lesson builds on 4 skills from earlier math. Start with “Find a unit fraction of a whole number by dividing”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Find a unit fraction of a whole number by dividingGr 4
2Multiply a fraction by a whole numberGr 5
3Predict whether a fraction of a number is smaller or larger than the numberGr 5
4Decompose a decimal into place-value parts that rebuild it exactlyGr 5
2-Minute Quick Check
1/6 of 540? 5/6 of 540? Is 6/5 of 540 more or less than 540?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find 1/8 of 480.
✓ Answer: 60
Task 2: Find 3/8 of 480.
✓ Answer: 180
Task 3: Without calculating, is 4/3 of 270 more or less than 270? Explain how you know.
✓ Answer: More. 4/3 is greater than 1, so taking 4/3 of a number gives more than the whole number (4/3 of 270 is 360).
Task 4: Write 105.76 as a hundreds part plus the rest, and add your parts to check.
✓ Answer: 100 + 5.76 = 105.76. The 1 is in the hundreds place, so it is worth 100.
Lesson 1-3
Math is In My World
Tram Problems · Choosing Multiplication or Division · Tape Diagrams and Tables
5.NBT.B.7
Why this unlocks Lesson 1.3: This lesson builds on 4 skills from earlier math. Start with “Multiply a decimal by 10 and by 100”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a decimal by 10 and by 100Gr 5
2Multiply a decimal by a whole numberGr 5
3Divide a multiple of ten by a two-digit divisorGr 5
4Convert hours to minutesGr 4
2-Minute Quick Check
12.5 × 10? 12.5 × 4? 4,000 ÷ 80? How many minutes are in 10 hours?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 3.6 × 100.
✓ Answer: 360
Task 2: Compute 7.5 × 8.
✓ Answer: 60
Task 3: Compute 3,200 ÷ 80.
✓ Answer: 40
Task 4: How many minutes are in 6 hours?
✓ Answer: 360
Lesson 1-4
Math is Explaining and Sharing
Arguments with Equations and Words · Volume of Boxes · Comparing Total Volume
5.MD.C.5
Why this unlocks Lesson 1.4: This lesson builds on 4 skills from earlier math. Start with “Multiply three whole numbers in any order”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Add a list of numbers to a running total.
Fluency Needed First · 4 Prerequisite Skills
1Multiply three whole numbers in any orderGr 4
2Find the volume of a rectangular prism with V = l × w × hGr 5
3Multiply one box's volume by the number of identical boxesGr 4
4Add the volumes of two prisms to find a combined volumeGr 5
2-Minute Quick Check
8 × 3 × 10? A box is 10 in by 4 in by 10 in: its volume? Two of those boxes hold how much?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 5 × 4 × 6.
✓ Answer: 120
Task 2: A box is 7 cm long, 3 cm wide, and 5 cm tall. What is its volume in cubic centimeters?
✓ Answer: 105
Task 3: One box holds 240 cubic inches. How many cubic inches do 3 identical boxes hold?
✓ Answer: 720
Task 4: One bed holds 30 cubic feet of soil and another holds 18 cubic feet. How many cubic feet do both hold?
✓ Answer: 48
Lesson 1-5
Math is Finding Patterns
Coin Value Patterns · Pattern Rules and Tables of Values · Generalizations
5.OA.B.3
Why this unlocks Lesson 1.5: This lesson builds on 4 skills from earlier math. Start with “Multiply a decimal money value by 100”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a decimal money value by 100Gr 5
2Extend a pattern by repeatedly applying an add ruleGr 4
3Find a later term as start + number of steps × ruleGr 5
4Recognize multiples of 5 by their last digitGr 4
2-Minute Quick Check
100 × $0.25? Start at 6 and add 3 four times. Start at 18 and add 2 ten times.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Each dime is worth $0.10. How many dollars are 100 dimes worth? Enter the number only.
✓ Answer: 10
Task 2: Start at 9 and apply the rule add 4 to write the next four terms.
✓ Answer: 9, 13, 17, 21, 25.
Task 3: A pattern starts at 15 and the rule is add 6. What is the value after 8 steps?
✓ Answer: 63
Task 4: Which of these could be the number of pennies in a jar worth a whole number of nickels: 132, 145, 158? Explain using the last digit.
✓ Answer: 145, because multiples of 5 end in 0 or 5. 132 and 158 do not end in 0 or 5.
Lesson 1-6
Math is Ours
The Problem-Solving Process · Getting Unstuck · Times-as-Many Problems
MP.3
Why this unlocks Lesson 1.6: This lesson builds on 4 skills from earlier math. Start with “Multiply to find the total in equal groups”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Multiply to find the total in equal groupsGr 3
2Interpret “times as many” as multiplication, not additionGr 4
3Multiply a two-digit number by a one-digit numberGr 4
4Divide to find the number of equal groupsGr 3
2-Minute Quick Check
8 × 2? 6 times as many as 16? How many bikes have 96 wheels if each bike has 2 wheels?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: A rack holds 9 bikes with 2 wheels each. How many wheels?
✓ Answer: 18
Task 2: A student says 5 times as many as 12 is 17. Explain the error and give the correct value.
✓ Answer: The student added 5 + 12. Times as many means 5 copies of 12: 12 × 5 = 60.
Task 3: Compute 18 × 4.
✓ Answer: 72
Task 4: A rack has 84 wheels and every bike has 2 wheels. How many bikes are on the rack?
✓ Answer: 42
Unit 212 lessons
Unit 2: Statistics and Data Displays
Order a list of numbers, read a scaled number line, and add then divide a set of values. A student who cannot order numbers cannot find a median, a quartile, or an outlier.
Lesson 2-1
Understand Statistical Questions
Statistical Questions · Dot Plots and Shape
6.SP.A.1
Why this unlocks Lesson 2.1: This lesson builds on 4 skills from earlier math. Start with “Order a list of up to 10 whole numbers least to greatest”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Order a list of numbers least to greatest.
Fluency Needed First · 4 Prerequisite Skills
1Order a list of up to 10 whole numbers least to greatestGr 4
2Read a scaled number line and say what one tick mark is worthGr 4
3Tally and count a set of up to 30 responses without losing placeGr 3
4Compare two whole numbers and say which is greaterGr 2
2-Minute Quick Check
Order 7, 3, 12, 5, 9. On a line from 0 to 10 split into 5 equal jumps, what is one jump worth?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Order 18, 7, 12, 7, 25 from least to greatest.
✓ Answer: 7, 7, 12, 18, 25
Task 2: A number line goes from 0 to 30 in 6 equal jumps. How much is one jump?
✓ Answer: 5
Task 3: A survey records 4 yes responses, then 5 yes responses, then 3 yes responses. How many yes responses?
✓ Answer: 12
Task 4: Which is greater: 47 or 74?
✓ Answer: 74
Lesson 2-2
Represent and Describe Data in a Histogram
Building a Histogram · Describing Histogram Shape
6.SP.B.4
Why this unlocks Lesson 2.2: This lesson builds on 4 skills from earlier math. Start with “Decide which interval a value belongs to (is 19 in 10–19 or 20–29?)”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total.
Fluency Needed First · 4 Prerequisite Skills
1Decide which interval a value belongs to (is 19 in 10–19 or 20–29?)Gr 5
2Sort up to 20 values into given intervals and count each binGr 5
3Add a column of one- and two-digit numbers to a totalGr 4
4Read a graph scale that skip-counts by 2, 5, or 10Gr 4
2-Minute Quick Check
Which interval holds 29: 20–29 or 30–39? How many of 4, 11, 19, 21, 25 land in 10–19?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Which interval contains 29: 10–19, 20–29, or 30–39?
✓ Answer: 20–29
Task 2: Sort 2, 8, 11, 14, 19, 22 into intervals 0–9, 10–19, 20–29. Give the three frequencies in order.
✓ Answer: 2, 3, 1
Task 3: Find the total frequency: 8 + 12 + 7 + 3.
✓ Answer: 30
Task 4: A frequency axis is labeled 0, 5, 10, 15. What value is two labeled steps above 0?
✓ Answer: 10
Lesson 2-3
Describe the Data Using the Median
Finding the Median · Median of an Even Set
6.SP.B.5.c
Why this unlocks Lesson 2.3: This lesson builds on 4 skills from earlier math. Start with “Order up to 12 numbers least to greatest”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Order a list of numbers least to greatest.
Fluency Needed First · 4 Prerequisite Skills
1Order up to 12 numbers least to greatestGr 4
2Find the middle position of an odd-length list by counting in from both endsGr 5
3Find the number halfway between two numbersGr 5
4Divide a two-digit sum by 2Gr 4
2-Minute Quick Check
Put in order: 9, 4, 15, 6, 11. What number is halfway between 12 and 17?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Order 15, 8, 12, 8, 17, 10.
✓ Answer: 8, 8, 10, 12, 15, 17
Task 2: Find the median of 3, 5, 8, 11, 14.
✓ Answer: 8
Task 3: What number is halfway between 18 and 26?
✓ Answer: 22
Task 4: Compute 46 ÷ 2.
✓ Answer: 23
Lesson 2-4
Represent and Describe Data in a Box Plot
Building a Box Plot · Comparing Box Plots
6.SP.B.4
Why this unlocks Lesson 2.4: This lesson builds on 4 skills from earlier math. Start with “Order a data set and find its median”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Order a list of numbers least to greatest.
Fluency Needed First · 4 Prerequisite Skills
1Order a data set and find its median2-3
2Split an ordered list into a lower half and an upper half2-3
3Subtract two-digit whole numbersGr 4
4Plot a point at the right place on a scaled number lineGr 5
2-Minute Quick Check
Ordered: 3, 5, 8, 10, 12, 14, 18, 20. Find the median of the lower half.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the median of 12, 4, 8, 6, 10.
✓ Answer: 8
Task 2: Split 2, 4, 6, 8, 10, 12 into equal lower and upper halves.
✓ Answer: Lower: 2, 4, 6; upper: 8, 10, 12
Task 3: Compute 73 − 28.
✓ Answer: 45
Task 4: A line has ticks at 0, 4, 8, 12, 16. Which value is halfway between 8 and 12?
✓ Answer: 10
Lesson 2-5
Describe Data by Range and Interquartile Range
Range and IQR · Comparing Variability
6.SP.B.5.c
Why this unlocks Lesson 2.5: This lesson builds on 4 skills from earlier math. Start with “Find the maximum and minimum of a data set”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Order a list of numbers least to greatest.
Fluency Needed First · 4 Prerequisite Skills
1Find the maximum and minimum of a data setGr 4
2Subtract whole numbers, including across zerosGr 4
3Locate Q1 and Q3 in an ordered set2-4
4Compare two differences to say which spread is largerGr 4
2-Minute Quick Check
Range of 12, 7, 19, 4, 15? If Q1 = 6 and Q3 = 17, what is Q3 − Q1?
Why this unlocks Lesson 2.6: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication and division facts through 12 × 12”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication and division facts through 12 × 12Gr 3
2Use place value to divide hundreds, tens, and onesGr 4
3Subtract multi-digit whole numbers with regroupingGr 4
4Estimate a quotient and check with multiplicationGr 5
2-Minute Quick Check
Compute 156 ÷ 6. Explain how multiplication checks your quotient.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 84 ÷ 7.
✓ Answer: 12
Task 2: Compute 248 ÷ 4.
✓ Answer: 62
Task 3: Compute 302 − 178.
✓ Answer: 124
Task 4: Estimate 238 ÷ 6 using the nearby compatible dividend 240.
✓ Answer: 40
Lesson 2-7
Divide Decimals Using an Algorithm
Dividing a Decimal by a Whole Number · Dividing by a Decimal
6.NS.B.3
Why this unlocks Lesson 2.7: This lesson builds on 4 skills from earlier math. Start with “Recall division facts through 12 × 12”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal.
Fluency Needed First · 4 Prerequisite Skills
1Recall division facts through 12 × 12Gr 3–4
2Divide multi-digit whole numbers with the standard algorithmGr 5
3Multiply and divide by 10, 100, 1000 by shifting the decimal pointGr 5
4Line up decimal places and annex zeros to keep dividingGr 5
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 96 ÷ 8.
✓ Answer: 12
Task 2: Compute 156 ÷ 6.
✓ Answer: 26
Task 3: Compute 4.8 × 100.
✓ Answer: 480
Task 4: Compute 7.5 ÷ 2.
✓ Answer: 3.75
Lesson 2-8
Describe Data Using the Mean
Finding the Mean · Reaching a Target Mean
6.SP.B.5.c
Why this unlocks Lesson 2.8: This lesson builds on 4 skills from earlier math. Start with “Add a list of 5–10 two-digit numbers accurately”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total.
Fluency Needed First · 4 Prerequisite Skills
1Add a list of 5–10 two-digit numbers accuratelyGr 4
2Divide a three-digit number by a one-digit numberGr 5
3Count how many values are in a data setGr 3
4Find a missing addend that reaches a given totalGr 4
2-Minute Quick Check
Mean of 10, 14, 8, 12, 16? What must be added to 48 to reach 60?
Why this unlocks Lesson 2.9: This lesson builds on 4 skills from earlier math. Start with “Find the mean of a data set”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total.
Fluency Needed First · 4 Prerequisite Skills
1Find the mean of a data set2-8
2Subtract to find the distance between two numbers and keep it positiveGr 5
3Add a list of distancesGr 4
4Divide a total by the number of values2-8
2-Minute Quick Check
The mean is 20. How far is 26 from the mean? How far is 14?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the mean of 4, 6, 8, 10.
✓ Answer: 7
Task 2: What is the distance between 3 and 11 on a number line?
✓ Answer: 8
Task 3: Add the distances 3, 1, 1, 3.
✓ Answer: 8
Task 4: Eight units of total deviation are shared across four values. What is 8 ÷ 4?
✓ Answer: 2
Lesson 2-10
Choose Appropriate Measures
Mean or Median? · Matching Center and Spread
6.SP.B.5.d
Why this unlocks Lesson 2.10: This lesson builds on 4 skills from earlier math. Start with “Compute mean, median, range, IQR, and MAD on demand”, then the rest, so lesson time goes to the new idea.
Fluency Needed First · 4 Prerequisite Skills
1Compute mean, median, range, IQR, and MAD on demand2-3, 2-5, 2-8, 2-9
2Spot an outlier in a short list2-1, 2-2
3Read a dot plot or histogram as symmetric or skewed2-1, 2-2
4Compare two measures and say which is pulled by an extreme value2-8
2-Minute Quick Check
Which measure of center fits 2, 3, 3, 4, 50 — mean or median? Why?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: For 2, 4, 6, 8, find mean, median, range, IQR, and MAD. Use medians of the two halves for quartiles.
✓ Answer: Mean 5; median 5; range 6; IQR 4; MAD 2
Task 2: Which value is a likely outlier: 8, 9, 10, 11, 52?
✓ Answer: 52
Task 3: A dot plot has 1 dot at 1, 2 dots at 2, 3 dots at 3, 2 dots at 4, and 1 dot at 5. Is it symmetric or skewed?
✓ Answer: Symmetric
Task 4: Replace 6 with 60 in 2, 4, 6. Which changes more: mean or median?
✓ Answer: Mean
Lesson 2-11
Add and Subtract Decimals
Line Up Decimal Points · Annex Zeros · Regroup Across the Decimal Point
6.NS.B.3
Why this unlocks Lesson 2.11: This lesson builds on 4 skills from earlier math. Start with “Name the place value of each digit in a decimal through thousandths”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total.
Fluency Needed First · 4 Prerequisite Skills
1Name the place value of each digit in a decimal through thousandthsGr 5
2Annex zeros so two decimals have the same number of decimal placesGr 5
3Add multi-digit whole numbers with regroupingGr 4
4Subtract multi-digit whole numbers by regrouping across zerosGr 4
2-Minute Quick Check
In 27.64, what is the value of the 4? Rewrite 9.6 with two decimal places. 1,000 − 473?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: In 6.385, what is the value of the 8? Enter the value only.
✓ Answer: 0.08
Task 2: Annex a zero to 3.7 so it has two decimal places, then find 3.70 + 2.05.
✓ Answer: 5.75
Task 3: Compute 2,764 + 1,548.
✓ Answer: 4312
Task 4: Compute 3,000 − 1,627.
✓ Answer: 1373
Lesson 2-12
Multiply Decimals
Multiply as Whole Numbers · Count Decimal Places · Estimate to Place the Point
6.NS.B.3
Why this unlocks Lesson 2.12: This lesson builds on 4 skills from earlier math. Start with “Multiply a multi-digit whole number by a two-digit whole number”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide by 10, 100, and 1000 by shifting the decimal.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a multi-digit whole number by a two-digit whole numberGr 5
2Recall multiplication facts through 12 × 12Gr 3
3Round a decimal to the nearest whole number to estimateGr 5
4Line up decimal points to add and subtract decimals2-11
2-Minute Quick Check
34 × 26? 7 × 8? Round 4.8 to the nearest whole number.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 47 × 23.
✓ Answer: 1081
Task 2: Compute 9 × 12.
✓ Answer: 108
Task 3: Round 6.7 to the nearest whole number.
✓ Answer: 7
Task 4: Find 4.25 + 3.6.
✓ Answer: 7.85
Unit 310 lessons
Unit 3: Ratios and Rates
Multiplication and division facts, and scaling a pair of numbers by the same factor. Every ratio table, graph, comparison, and conversion in this unit is that one move repeated.
Lesson 3-1
Understand Ratios
Understand Ratios · Ratios with Tape Diagrams
6.RP.A.1
Why this unlocks Lesson 3.1: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication facts through 12 × 12”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication facts through 12 × 12Gr 3
2Skip-count by any one-digit numberGr 3
3Count each part of a part-whole picture, then add the parts for the totalGr 2–4
4Write a fraction naming a part of a wholeGr 4
2-Minute Quick Check
6 × 4? There are 3 dogs and 5 cats — how many animals in all?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 8 × 7.
✓ Answer: 56
Task 2: Continue the pattern: 6, 12, 18, __.
✓ Answer: 24
Task 3: There are 5 blue counters and 7 red counters. How many counters in all?
✓ Answer: 12
Task 4: Three of eight equal slices are shaded. Write the shaded fraction.
✓ Answer: 3/8
Lesson 3-2
Rates and Unit Rates
Understanding Unit Rates · Comparing Unit Prices (Best Buy)
6.RP.A.2
Why this unlocks Lesson 3.2: This lesson builds on 4 skills from earlier math. Start with “Divide a multi-digit number by a one- or two-digit number”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Divide a multi-digit number by a one- or two-digit numberGr 5
2Divide to find the amount for one, and read “per” as “for each one”Gr 4
3Keep a repeating pattern going in a tableGr 4
4Attach the correct unit label to a computed numberGr 4
2-Minute Quick Check
60 ÷ 4? If 4 pencils cost $2, what does 1 pencil cost?
Why this unlocks Lesson 3.3: This lesson builds on 4 skills from earlier math. Start with “Multiply both numbers of a pair by the same factor”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Multiply both numbers of a pair by the same factorGr 4
2Recall multiplication facts through 12 × 12Gr 3
3Write equivalent fractions by multiplying numerator and denominatorGr 4
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Scale the pair 3 : 5 by a factor of 4.
✓ Answer: 12 : 20
Task 2: Compute 9 × 8.
✓ Answer: 72
Task 3: Fill in the numerator: 2/3 = __/15.
✓ Answer: 10
Task 4: A table lists (2, 6), (4, 12), (6, 18). What pair comes next if the same step continues?
✓ Answer: (8, 24)
Lesson 3-4
Equivalent Ratios Using Graphs
Graphing Ratios on Coordinate Planes · Interpreting Points on Ratio Graphs
6.RP.A.3.a
Why this unlocks Lesson 3.4: This lesson builds on 4 skills from earlier math. Start with “Plot a point in Quadrant I from an ordered pair”, then the rest, so lesson time goes to the new idea. The core skill behind it: Scale a pair of numbers by the same factor; Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Plot a point in Quadrant I from an ordered pairGr 5
2Read the scale on each axis before plottingGr 5
3Count right from the origin, then upGr 5
4Build a ratio table to generate the pairs3-3
2-Minute Quick Check
Start at (0, 0), move 3 units right and 2 units up — write the ordered pair. A horizontal axis has grid lines every 5 units, starting at 0. How many intervals right of 0 is the value 15?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Describe how to plot (3, 5) from the origin.
✓ Answer: Move 3 units right and 5 units up
Task 2: The x-axis ticks are 0, 2, 4, 6, 8. What x-value is at the third tick to the right of 0?
✓ Answer: 6
Task 3: A point is 4 units right and 7 units up from the origin. Give its ordered pair.
✓ Answer: (4, 7)
Task 4: The rule is y = 3x. Give y when x = 4.
✓ Answer: 12
Lesson 3-5
Comparing Ratio Relationships
Comparing Ratios with Common Terms · Comparing Rates with Benchmarks
6.RP.A.3.b
Why this unlocks Lesson 3.5: This lesson builds on 4 skills from earlier math. Start with “Compare two whole numbers”, then the rest, so lesson time goes to the new idea. The core skill behind it: Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Compare two whole numbersGr 2
2Scale both parts of a ratio to hit a target value3-3
3Find a common multiple of two numbersGr 5
4Divide to find a unit rate for a fair comparison3-2
2-Minute Quick Check
Which is greater, 12 or 9? Scale 3 : 4 so the first number is 12.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Which is greater: 35 or 53?
✓ Answer: 53
Task 2: Scale 4 : 7 to an equivalent ratio with first term 20.
✓ Answer: 20 : 35
Task 3: What is the least common multiple of 6 and 8?
✓ Answer: 24
Task 4: Pack A costs $12 for 4 pens. Pack B costs $14 for 7 pens. Which has the lower price per pen?
✓ Answer: Pack B
Lesson 3-6
Converting Within One System
Customary Unit Conversions · Metric Unit Conversions
6.RP.A.3.d
Why this unlocks Lesson 3.6: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication and division facts”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide by 10, 100, and 1000 by shifting the decimal; Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication and division factsGr 3
2Recall customary facts: 1 ft = 12 in, 1 lb = 16 oz, 1 gal = 4 qtGr 4–5
3Multiply and divide by 10, 100, 1000 for metric stepsGr 5
4Decide whether a conversion calls for multiplying or dividingGr 5
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 72 ÷ 9.
✓ Answer: 8
Task 2: Use 1 foot = 12 inches. How many inches are in 3 feet?
✓ Answer: 36
Task 3: How many meters are in 2.4 kilometers? Use 1 km = 1000 m.
✓ Answer: 2400
Task 4: Convert 48 inches to feet using 12 inches = 1 foot.
✓ Answer: 4
Lesson 3-7
Converting Between Systems
Customary vs. Metric Conversions · Multi-Step System Conversions and Dosage
6.RP.A.3.d
Why this unlocks Lesson 3.7: This lesson builds on 4 skills from earlier math. Start with “Multiply a whole number by a decimal”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a whole number by a decimalGr 5
2Round a decimal to the nearest tenth or hundredthGr 5
3Multiply by a conversion factor and keep the unit label3-6
4Chain two conversions in order without losing a step3-6
2-Minute Quick Check
4 × 1.6? 10 × 0.45? Round 6.44 to the nearest tenth.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 6 × 2.5.
✓ Answer: 15
Task 2: Round 7.846 to the nearest hundredth.
✓ Answer: 7.85
Task 3: Use 1 inch = 2.54 centimeters. Convert 5 inches to centimeters.
✓ Answer: 12.7
Task 4: Use 1 yard = 3 feet and 1 foot = 12 inches. Convert 2 yards to inches.
✓ Answer: 72
Lesson 3-8
Solve Problems with Unit Rates
Find a Unit Price · Compare Deals · Use a Unit Rate to Find a Total
6.RP.A.3.b
Why this unlocks Lesson 3.8: This lesson builds on 4 skills from earlier math. Start with “Divide a money amount by a whole number”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Scale a pair of numbers by the same factor.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute $7.20 ÷ 6. Enter the value only.
✓ Answer: 1.2
Task 2: Which is greater, 0.4 or 0.35? Enter the greater number.
✓ Answer: 0.4
Task 3: Compute 8 × 0.65.
✓ Answer: 5.2
Task 4: A cyclist rides 36 miles in 3 hours. How many miles per hour is that?
✓ Answer: 12
Lesson 3-9
Equivalent Ratios
Scale Both Terms by the Same Factor · Simplify with the GCF · Check Equivalence
6.RP.A.3.a
Why this unlocks Lesson 3.9: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication and division facts through 12 × 12”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication and division facts through 12 × 12Gr 3
2Find the greatest common factor of two numbers within 100Gr 4
3Find a missing factor: 5 × ? = 20Gr 3
4Write a ratio in the order a:b and name what each term counts3-1
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 7 × 6.
✓ Answer: 42
Task 2: Find the greatest common factor of 18 and 24.
✓ Answer: 6
Task 3: Find the missing factor: 9 × ? = 63.
✓ Answer: 7
Task 4: A class has 12 girls and 15 boys. Write the ratio of boys to girls and explain which term comes first.
✓ Answer: 15:12, boys first, because the ratio names boys before girls.
Lesson 3-10
Convert Measurement Units
Multiply to a Smaller Unit · Divide to a Larger Unit · Mixed Units
6.RP.A.3.d
Why this unlocks Lesson 3.10: This lesson builds on 4 skills from earlier math. Start with “Recall common customary relationships: 12 in = 1 ft, 3 ft = 1 yd, 16 oz = 1 lb, 4 qt = 1 gal”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Recall common customary relationships: 12 in = 1 ft, 3 ft = 1 yd, 16 oz = 1 lb, 4 qt = 1 galGr 4
2Multiply a whole number or decimal by 10, 100, or 1,000Gr 5
3Divide with a remainder and interpret it in a measurementGr 4
4Use an equivalent ratio to find a missing value3-9
2-Minute Quick Check
How many ounces in 1 pound? 2.5 × 1,000? 30 ÷ 12 = ? remainder ?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: How many ounces are in 3 pounds? (1 lb = 16 oz)
✓ Answer: 48
Task 2: Compute 4.2 × 1,000.
✓ Answer: 4200
Task 3: 45 inches is how many whole feet, with some inches left over? Enter the number of whole feet.
✓ Answer: 3
Task 4: The ratios 1:3 and 7:? are equivalent. Find the missing term.
✓ Answer: 21
Unit 45 lessons
Unit 4: Percent
Move fluently among a fraction, a decimal, and a percent, and multiply or divide by 100 by shifting the decimal point. Benchmark percents (10%, 1%, 25%, 50%) do most of the work.
Lesson 4-1
Understand Percent
Percent Means Per Hundred · Percents Greater Than 100%
6.RP.A.3.c
Why this unlocks Lesson 4.1: This lesson builds on 4 skills from earlier math. Start with “Read and write a fraction with denominator 100”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Move among a fraction, a decimal, and a percent.
Fluency Needed First · 4 Prerequisite Skills
1Read and write a fraction with denominator 100Gr 4
2Count shaded squares on a 100-square gridGr 4
3Write hundredths as a decimalGr 5
4Simplify a fraction over 100Gr 4
2-Minute Quick Check
37 of 100 squares are shaded — write it as a fraction, a decimal, and a percent.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write thirty-seven hundredths as a fraction.
✓ Answer: 37/100
Task 2: A 10-by-10 grid has 4 complete rows shaded. How many squares are shaded?
✓ Answer: 40
Task 3: Write 68/100 as a decimal.
✓ Answer: 0.68
Task 4: Simplify 45/100.
✓ Answer: 9/20
Lesson 4-2
Relate Fractions, Decimals, and Percentages
Fractions, Decimals, Percents · Compare and Order
6.RP.A.3.c
Why this unlocks Lesson 4.2: This lesson builds on 4 skills from earlier math. Start with “Multiply and divide by 100 by shifting the decimal point”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Order a list of numbers least to greatest; Move among a fraction, a decimal, and a percent.
Fluency Needed First · 4 Prerequisite Skills
1Multiply and divide by 100 by shifting the decimal pointGr 5
2Write a fraction as a decimal by dividingGr 5
3Recall benchmark equivalents for 1/2, 1/4, 1/5, 3/4, 1/10Gr 4–5
4Compare decimals to the hundredths placeGr 5
2-Minute Quick Check
0.45 as a percent? 25% as a fraction in simplest form? Which is greater, 0.6 or 55%?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 0.36 × 100.
✓ Answer: 36
Task 2: Write 3/8 as a decimal.
✓ Answer: 0.375
Task 3: Write 3/4 as a percent.
✓ Answer: 75%
Task 4: Which is greater: 0.6 or 0.56?
✓ Answer: 0.6
Lesson 4-3
Estimate the Percent of a Number
Estimate with Benchmarks · Estimate a Percent of a Number
6.RP.A.3.c
Why this unlocks Lesson 4.3: This lesson builds on 4 skills from earlier math. Start with “Find 10% of a number by shifting the decimal one place left”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide by 10, 100, and 1000 by shifting the decimal; Move among a fraction, a decimal, and a percent.
Fluency Needed First · 4 Prerequisite Skills
1Find 10% of a number by shifting the decimal one place leftGr 5
2Find 1% of a number by shifting two places leftGr 5
3Halve and double a number mentallyGr 4
4Round to a compatible number that is easy to take a percent ofGr 5
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find 10% of 350.
✓ Answer: 35
Task 2: Find 1% of 800.
✓ Answer: 8
Task 3: Find half of 86.
✓ Answer: 43
Task 4: Estimate 25% of 198 by using 200 as a compatible number.
✓ Answer: 50
Lesson 4-4
Find and Compare with Percentages
Find the Percent of a Number · Compare with Percents
6.RP.A.3.c
Why this unlocks Lesson 4.4: This lesson builds on 4 skills from earlier math. Start with “Write a percent as a decimal”, then the rest, so lesson time goes to the new idea. The core skill behind it: Move among a fraction, a decimal, and a percent; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Write a percent as a decimal4-2
2Multiply a whole number by a decimalGr 5
3Multiply a whole number by a fractionGr 5
4Compare two products to decide which is largerGr 4
2-Minute Quick Check
0.25 × 80? 3/4 of 80? Which is more: 20% of 50 or 30% of 40?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write 18% as a decimal.
✓ Answer: 0.18
Task 2: Compute 60 × 0.35.
✓ Answer: 21
Task 3: Find 3/4 of 28.
✓ Answer: 21
Task 4: Which is greater: 20% of 90 or 30% of 70?
✓ Answer: 30% of 70
Lesson 4-5
Determine the Whole Given the Part and Percent
Find the Whole · Solve for the Whole
6.RP.A.3.c
Why this unlocks Lesson 4.5: This lesson builds on 4 skills from earlier math. Start with “Divide a whole number by a decimal”, then the rest, so lesson time goes to the new idea. The core skill behind it: Move among a fraction, a decimal, and a percent; Name the inverse of an operation and check by substituting back; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Divide a whole number by a decimal2-7
2Tell which number is the part and which is the whole4-4
3Divide by a benchmark fraction such as 1/4 or 1/2Gr 5
4Check an answer by multiplying backGr 4
2-Minute Quick Check
12 ÷ 0.25? 12 is 25% of what number? Now check: 25% of your answer.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 18 ÷ 0.3.
✓ Answer: 60
Task 2: Twelve students out of a class of 30 walk to school. Identify the part and the whole.
✓ Answer: Part 12; whole 30
Task 3: Compute 9 ÷ 1/4.
✓ Answer: 36
Task 4: A student says the whole is 80 when 25% is 20. Check by finding 25% of 80.
✓ Answer: 20
Unit 510 lessons
Unit 5: Area, Volume, and Surface Area
Multiply accurately, find the area of a rectangle and a triangle, and tell a perpendicular height from a slanted side. Every later formula in the unit is built from those two areas.
Lesson 5-1
Determine the Area of Parallelograms and Rhombuses
Area of Parallelograms · Area of Rhombuses
6.G.A.1
Why this unlocks Lesson 5.1: This lesson builds on 4 skills from earlier math. Start with “Multiply two- and three-digit numbers”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Multiply two- and three-digit numbersGr 5
2Find the area of a rectangle by multiplying length by widthGr 3–4
3Tell a base from a slanted side and identify the perpendicular heightGr 5
4Label an area answer in square unitsGr 4
2-Minute Quick Check
Area of an 8 cm by 5 cm rectangle? In a parallelogram, is the height the slanted side?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 24 × 13.
✓ Answer: 312
Task 2: A rectangle is 8 cm long and 6 cm wide. Find its area in square centimeters.
✓ Answer: 48
Task 3: A parallelogram has base 10 cm, slanted side 7 cm, and perpendicular height 4 cm. Which number is the height?
✓ Answer: 4
Task 4: A rectangle measures 3 meters by 5 meters. State its area with units.
✓ Answer: 15 square meters
Lesson 5-2
Determine the Area of Triangles
Area of Triangles · Missing Measures in Triangles
6.G.A.1
Why this unlocks Lesson 5.2: This lesson builds on 4 skills from earlier math. Start with “Multiply, then halve — or halve, then multiply”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Find the area of a rectangle and of a triangle; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Multiply, then halve — or halve, then multiplyGr 5
2Multiply a whole number by 1/2Gr 5
3Find the area of a rectangle or parallelogram5-1
4Identify the height drawn perpendicular to the chosen base5-1
2-Minute Quick Check
7 × 4, then half? Area of a triangle with base 7 m and height 4 m?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute (14 × 9) ÷ 2.
✓ Answer: 63
Task 2: Compute 1/2 × 38.
✓ Answer: 19
Task 3: A parallelogram has base 12 cm and height 5 cm. Find its area in square centimeters.
✓ Answer: 60
Task 4: A triangle has base 8 cm, perpendicular height 6 cm, and slanted side 10 cm. Which length belongs in the height part of its area formula?
✓ Answer: 6
Lesson 5-3
Determine the Area of Trapezoids
Trapezoids by Decomposing · The Trapezoid Area Formula
6.G.A.1
Why this unlocks Lesson 5.3: This lesson builds on 4 skills from earlier math. Start with “Add the two numbers inside parentheses before multiplying”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents; Find the area of a rectangle and of a triangle; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Add the two numbers inside parentheses before multiplyingGr 5
2Multiply by 1/25-2
3Follow order of operations inside a formulaGr 5
4Split a figure into a rectangle plus triangles5-2
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute (7 + 11) × 4.
✓ Answer: 72
Task 2: Compute 1/2 × 54.
✓ Answer: 27
Task 3: Evaluate 1/2 × (6 + 10) × 5.
✓ Answer: 40
Task 4: A shape is made of a rectangle of area 24 and two triangles of area 6 each, without overlap. Find the total area.
✓ Answer: 36
Lesson 5-4
Apply Area Concepts to Solve Problems
Area of Regular Polygons · Composite Figures
6.G.A.1
Why this unlocks Lesson 5.4: This lesson builds on 4 skills from earlier math. Start with “Find the area of rectangles, triangles, and trapezoids on demand”, then the rest, so lesson time goes to the new idea. The core skill behind it: Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Find the area of rectangles, triangles, and trapezoids on demand5-1, 5-2, 5-3
2Add and subtract areasGr 4
3Find a missing side length from the other labeled sidesGr 5
4Multiply one part's area by the number of identical partsGr 4
2-Minute Quick Check
A 10 by 6 rectangle has a 3 by 2 rectangle cut out. What area is left?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the area of a triangle with base 10 cm and height 8 cm, in square centimeters.
✓ Answer: 40
Task 2: A rectangle has area 72 square meters. A 15-square-meter section is removed. What area remains?
✓ Answer: 57
Task 3: A total side length is 14 cm. One part is 9 cm. How long is the other part?
✓ Answer: 5
Task 4: Six identical tiles each cover 12 square centimeters. What is the total area?
✓ Answer: 72
Lesson 5-5
Determine the Volume of Rectangular Prisms
Volume of Rectangular Prisms · Missing and Fractional Edges
6.G.A.2
Why this unlocks Lesson 5.5: This lesson builds on 4 skills from earlier math. Start with “Multiply three factors in any order”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Multiply three factors in any orderGr 5
2Multiply fractions and mixed numbersGr 5
3Find the area of a rectangular base5-1
4Label a volume answer in cubic unitsGr 5
2-Minute Quick Check
4 × 3 × 2? 1/2 × 1/3? Volume of a 4 by 3 by 2 prism?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 4 × 7 × 5.
✓ Answer: 140
Task 2: Compute 1 1/2 × 2/3.
✓ Answer: 1
Task 3: A rectangular base is 9 cm by 4 cm. Find its area in square centimeters.
✓ Answer: 36
Task 4: A prism measures 2 cm by 3 cm by 5 cm. State its volume with units.
✓ Answer: 30 cubic centimeters
Lesson 5-6
Represent Three-Dimensional Figures in Two Dimensions
Drawing Nets of Solids · Reading Nets and Solids
6.G.A.4
Why this unlocks Lesson 5.6: This lesson builds on 4 skills from earlier math. Start with “Name the faces, edges, and vertices of a prism and a pyramid”, then the rest, so lesson time goes to the new idea. The core skill behind it: Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Name the faces, edges, and vertices of a prism and a pyramidGr 5
2Count the faces of a solidGr 5
3Find the area of each rectangle and triangle inside a net5-1, 5-2
4Match congruent faces in pairsGr 4
2-Minute Quick Check
How many faces does a rectangular prism have? How many matching pairs is that?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: How many faces, edges, and vertices does a rectangular prism have?
✓ Answer: 6 faces, 12 edges, 8 vertices
Task 2: How many faces does a square pyramid have, including its base?
✓ Answer: 5
Task 3: A net has a rectangular face 3 cm by 7 cm. Find that face's area in square centimeters.
✓ Answer: 21
Task 4: A prism has face pairs sized 3 by 4, 3 by 5, and 4 by 5. How many faces are 3 by 4?
✓ Answer: 2
Lesson 5-7
Determine Surface Area of Prisms
Surface Area of Rectangular Prisms · Surface Area of Triangular Prisms
6.G.A.4
Why this unlocks Lesson 5.7: This lesson builds on 4 skills from earlier math. Start with “Add a list of five or six areas”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total; Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Add a list of five or six areasGr 4
2Find the area of rectangles and triangles quickly5-1, 5-2
3Read a net and pair up matching faces5-6
4Double one face's area to cover a matching pairGr 4
2-Minute Quick Check
Faces measure 12, 12, 20, 20, 15, 15 square units — what is the surface area?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Add the face areas 12 + 12 + 15 + 15 + 20 + 20.
✓ Answer: 94
Task 2: Find the area of a triangle with base 6 cm and height 9 cm, in square centimeters.
✓ Answer: 27
Task 3: In a rectangular prism with dimensions 2, 3, 5, list the three different face areas.
✓ Answer: 6, 10, 15
Task 4: One face has area 18 square centimeters. What is the combined area of it and its matching opposite face?
✓ Answer: 36
Lesson 5-8
Determine Surface Area of Pyramids
Surface Area of Pyramids · Lateral Area of Pyramids
6.G.A.4
Why this unlocks Lesson 5.8: This lesson builds on 4 skills from earlier math. Start with “Find the area of a triangle”, then the rest, so lesson time goes to the new idea. The core skill behind it: Add a list of numbers to a running total; Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Find the area of a triangle5-2
2Find the area of a square or rectangular base5-1
3Multiply one triangular face's area by the number of identical facesGr 4
4Tell slant height from vertical height on a net or drawing5-6
2-Minute Quick Check
One triangular face is 24 square inches. With four identical faces, what is the lateral area?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the area of a triangle with base 8 cm and perpendicular height 7 cm, in square centimeters.
✓ Answer: 28
Task 2: Find the area of a square base with side 6 cm, in square centimeters.
✓ Answer: 36
Task 3: Four congruent triangular faces each have area 21 square centimeters. Find their combined area.
✓ Answer: 84
Task 4: A pyramid's vertical height is 9 cm. The altitude drawn within a triangular face is 10 cm. Which height finds that face's area?
✓ Answer: 10
Lesson 5-9
Area of Regular Polygons
Decompose a Regular Polygon into Central Triangles · Estimate Area with Rounded Heights
6.G.A.1
Why this unlocks Lesson 5.9: This lesson builds on 4 skills from earlier math. Start with “Find the area of a triangle with A = ½ × base × height”, then the rest, so lesson time goes to the new idea. The core skill behind it: Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Find the area of a triangle with A = ½ × base × height5-2
2Multiply a decimal by a whole numberGr 5
3Name how many sides a pentagon, hexagon, and octagon haveGr 3
4Find half of a whole number or a decimalGr 5
2-Minute Quick Check
½ × 6 × 5.2? 6 × 15.6? How many sides does an octagon have?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the area of a triangle with base 9 cm and height 4 cm.
✓ Answer: 18
Task 2: Compute 8 × 3.6.
✓ Answer: 28.8
Task 3: How many central triangles does a regular pentagon, a regular hexagon, and a regular octagon each make? Explain the pattern.
✓ Answer: A pentagon makes 5, a hexagon makes 6, and an octagon makes 8: one central triangle for each side.
Task 4: Find half of 31.2.
✓ Answer: 15.6
Lesson 5-10
Volume of Rectangular Prisms
Volume with Fractional Edge Lengths · Base Area × Height · Missing Edge from Volume
6.G.A.2
Why this unlocks Lesson 5.10: This lesson builds on 4 skills from earlier math. Start with “Rewrite a mixed number as an improper fraction”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide with fractions and mixed numbers; Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Rewrite a mixed number as an improper fractionGr 5
2Multiply a fraction by a whole numberGr 5
3Multiply a decimal by whole numbersGr 5
4Find a missing factor by dividing the product by the known factors5-5
2-Minute Quick Check
Write 2 1/2 as an improper fraction. 3/4 × 8? 36 ÷ (6 × 3)?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write 3 3/4 as an improper fraction.
✓ Answer: 15/4
Task 2: Compute 2/3 × 9.
✓ Answer: 6
Task 3: Compute 4 × 2.5 × 3.
✓ Answer: 30
Task 4: A box has a volume of 48 cubic inches, a length of 4 inches, and a width of 3 inches. What is its height in inches?
✓ Answer: 4
Unit 615 lessons
Unit 6: Expressions
Order of operations, and the distributive property in both directions. Fraction division opens the unit, so reciprocals and improper fractions must be automatic first.
Lesson 6-1
Division Expressions with Fractions and Whole Numbers
Dividing by a Unit Fraction · Dividing a Fraction by a Whole Number
6.NS.A.1
Why this unlocks Lesson 6.1: This lesson builds on 4 skills from earlier math. Start with “Divide decimals and whole numbers”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Divide decimals and whole numbers2-7
2Name the reciprocal of a whole number and of a unit fractionGr 5
3Multiply a whole number by a fractionGr 5
4Read a fraction bar as “divided by”Gr 5
2-Minute Quick Check
4.8 ÷ 2? How many fourths are in 3? Reciprocal of 5?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 8.4 ÷ 4.
✓ Answer: 2.1
Task 2: What is the reciprocal of 5?
✓ Answer: 1/5
Task 3: Compute 12 × 2/3.
✓ Answer: 8
Task 4: Write 15/3 as a division expression and evaluate it.
✓ Answer: 15 ÷ 3 = 5
Lesson 6-2
Division Expressions with Fractions and Mixed Numbers
Dividing a Fraction by a Fraction · Dividing with Mixed Numbers
6.NS.A.1
Why this unlocks Lesson 6.2: This lesson builds on 4 skills from earlier math. Start with “Rewrite a mixed number as an improper fraction”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Rewrite a mixed number as an improper fractionGr 5
2Multiply two fractionsGr 5
3Find the reciprocal of any fraction6-1
4Simplify a fraction to lowest termsGr 4
2-Minute Quick Check
2 3/4 as an improper fraction? Reciprocal of 3/5? 2/3 × 5/4?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write 2 3/4 as an improper fraction.
✓ Answer: 11/4
Task 2: Compute 2/3 × 3/5.
✓ Answer: 2/5
Task 3: Find the reciprocal of 4/7.
✓ Answer: 7/4
Task 4: Simplify 18/24.
✓ Answer: 3/4
Lesson 6-3
Explore Numerical Expressions with Exponents
Writing Repeated Factors as Powers · Finding the Value of a Power
6.EE.A.1
Why this unlocks Lesson 6.3: This lesson builds on 4 skills from earlier math. Start with “Multiply the same factor repeatedly”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Multiply and divide by 10, 100, and 1000 by shifting the decimal.
Fluency Needed First · 4 Prerequisite Skills
1Multiply the same factor repeatedlyGr 4
2Recall 2², 3², 4², 5², 2³, and 3³Gr 5–6
3Multiply by 10 repeatedly to build powers of 10Gr 5
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 3 × 3 × 3 × 3.
✓ Answer: 81
Task 2: Evaluate 5².
✓ Answer: 25
Task 3: Evaluate 10 × 10 × 10.
✓ Answer: 1000
Task 4: How many factors of 2 appear in 2 × 2 × 2 × 2 × 2?
✓ Answer: 5
Lesson 6-4
Write and Evaluate Numerical Expressions with Exponents
Writing Numerical Expressions · Order of Operations with Powers
6.EE.A.2.c
Why this unlocks Lesson 6.4: This lesson builds on 4 skills from earlier math. Start with “Evaluate a power”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents.
Fluency Needed First · 4 Prerequisite Skills
1Evaluate a power6-3
2Follow order of operations with parentheses and exponentsGr 5
3Read a fraction bar as a grouping symbolGr 5
4Translate “sum,” “product,” and “less than” into symbolsGr 5
2-Minute Quick Check
4 + 6 ÷ 2? 3 + 2² × 5? Which operation comes first?
Why this unlocks Lesson 6.5: This lesson builds on 4 skills from earlier math. Start with “Substitute a number for a letter, then compute”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents.
Fluency Needed First · 4 Prerequisite Skills
1Substitute a number for a letter, then computeGr 6
2Follow order of operations after substituting6-4
3Read 4n as 4 × nGr 6
4Translate a word phrase into an expression6-4
2-Minute Quick Check
Evaluate 4n + 3 when n = 5. Write an expression for “7 more than a number.”
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Evaluate n + 8 when n = 6.
✓ Answer: 14
Task 2: Evaluate 2x + 3 when x = 5.
✓ Answer: 13
Task 3: Evaluate 4n when n = 7.
✓ Answer: 28
Task 4: Write an expression for 9 more than a number n.
✓ Answer: n + 9
Lesson 6-6
Identify Equivalent Algebraic Expressions
Simplifying to Show Equivalence · Testing Equivalence by Substituting
6.EE.A.4
Why this unlocks Lesson 6.6: This lesson builds on 4 skills from earlier math. Start with “Combine like terms such as 5x + 2x”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents.
Fluency Needed First · 4 Prerequisite Skills
1Combine like terms such as 5x + 2xGr 6
2Evaluate an expression at a chosen value6-5
3Apply the distributive property with numbers, e.g. 3(4 + 5)Gr 5
4Compare two results to decide whether expressions matchGr 4
2-Minute Quick Check
5x + 2x = ? Evaluate 2(x + 3) and 2x + 6 at x = 4 — are they the same?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Simplify 5x + 2x.
✓ Answer: 7x
Task 2: Evaluate 3x + 4 when x = 2.
✓ Answer: 10
Task 3: Evaluate 3(4 + 5) using distribution.
✓ Answer: 27
Task 4: Compare 2(x + 3) and 2x + 6 at x = 4. Give both values.
✓ Answer: 14 and 14
Lesson 6-7
Find Factors and Multiples
Factors and the Greatest Common Factor · Multiples and the Least Common Multiple
6.NS.B.4
Why this unlocks Lesson 6.7: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication and division facts”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication and division factsGr 3
2List every factor pair of a number up to 100Gr 4
3List the first several multiples of a numberGr 4
4Test divisibility by 2, 3, 5, 9, and 10Gr 5
2-Minute Quick Check
All the factors of 18? First five multiples of 6? GCF of 18 and 24?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 84 ÷ 7.
✓ Answer: 12
Task 2: List every positive factor of 18.
✓ Answer: 1, 2, 3, 6, 9, 18
Task 3: List the first four positive multiples of 7.
✓ Answer: 7, 14, 21, 28
Task 4: Which of 2, 3, 5, 9, and 10 divide 135 evenly?
✓ Answer: 3, 5, 9
Lesson 6-8
Generate Equivalent Expressions
Using Properties to Rewrite Expressions · Expanding and Factoring Expressions
6.EE.A.3
Why this unlocks Lesson 6.8: This lesson builds on 4 skills from earlier math. Start with “Find the GCF of two numbers”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents.
Fluency Needed First · 4 Prerequisite Skills
1Find the GCF of two numbers6-7
2Apply the distributive property forward and backward6-6
3Combine like terms6-6
4Multiply a number by a sumGr 5
2-Minute Quick Check
GCF of 18 and 24? Expand 3(x + 5). Factor 4x + 12.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the greatest common factor of 18 and 30.
✓ Answer: 6
Task 2: Factor 12x + 20 using the greatest common factor.
✓ Answer: 4(3x + 5)
Task 3: Simplify 4a + 3a + 2.
✓ Answer: 7a + 2
Task 4: Compute 6(10 + 4).
✓ Answer: 84
Lesson 6-9
Divide Whole Numbers by Fractions
How Many Groups of a Fraction Fit in a Whole Number · Multiply by the Reciprocal · Leftover Parts of a Group
6.NS.A.1
Why this unlocks Lesson 6.9: This lesson builds on 4 skills from earlier math. Start with “Count how many unit fractions make one whole”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide with fractions and mixed numbers; Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Count how many unit fractions make one wholeGr 5
2Multiply a whole number by a fractionGr 5
3Write the reciprocal of a fraction or whole number6-1
4Interpret a leftover part of a group in a word problemGr 4
2-Minute Quick Check
How many 1/4s are in 1? Reciprocal of 3/5? 6 × 3/4?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: How many 1/5s are in 3?
✓ Answer: 15
Task 2: Compute 9 × 2/3.
✓ Answer: 6
Task 3: Write the reciprocal of 2/7.
✓ Answer: 7/2
Task 4: There are 5 cups of trail mix. Each bag holds 2/3 cup. How many full bags can be filled, and what is left over? Explain.
✓ Answer: 5 ÷ 2/3 = 7 1/2, so 7 full bags can be filled with half a bag's amount (1/3 cup) left over.
Lesson 6-10
Divide Mixed Numbers
Rewrite Mixed Numbers as Improper Fractions · Keep, Change, Flip · Write the Quotient as a Mixed or Whole Number
6.NS.A.1
Why this unlocks Lesson 6.10: This lesson builds on 4 skills from earlier math. Start with “Rewrite a mixed number as an improper fraction”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Rewrite a mixed number as an improper fractionGr 5
2Write the reciprocal of a fraction6-9
3Multiply two fractionsGr 5
4Rewrite an improper fraction as a whole or mixed numberGr 4
2-Minute Quick Check
Write 3 1/2 as an improper fraction. 7/2 × 4/1? Write 28/2 as a whole number.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write 4 2/3 as an improper fraction.
✓ Answer: 14/3
Task 2: Write the reciprocal of 5/8.
✓ Answer: 8/5
Task 3: Multiply 9/4 × 2/3.
✓ Answer: 3/2
Task 4: Write 22/4 as a mixed number.
✓ Answer: 5 1/2
Lesson 6-11
Fraction Division Problem Solving
Identify the Dividend and Divisor in a Story · Write and Solve the Division Equation · Check for Reasonableness
6.NS.A.1
Why this unlocks Lesson 6.11: This lesson builds on 4 skills from earlier math. Start with “Name the total (dividend) and the group size (divisor) in a story”, then the rest, so lesson time goes to the new idea. The core skill behind it: Multiply and divide with fractions and mixed numbers; Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Name the total (dividend) and the group size (divisor) in a story6-9
2Divide a fraction by a fraction using the reciprocal6-1
3Multiply a whole number by a fraction to check a quotientGr 5
4Compare a divisor to 1 to predict whether a quotient is greater or less than the dividendGr 5
2-Minute Quick Check
In 'How many 1/8-pound casts can be made from 7/8 pound?', which number is the dividend? 7/8 ÷ 1/8? 7 × 1/8?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: A jug holds 7/8 quart. Each cup holds 1/8 quart. Write the division equation and name the dividend and the divisor.
✓ Answer: 7/8 ÷ 1/8 = 7. The dividend is 7/8 quart (the total); the divisor is 1/8 quart (the size of each cup).
Task 2: Compute 5/6 ÷ 1/3.
✓ Answer: 5/2
Task 3: Compute 6 × 1/8.
✓ Answer: 3/4
Task 4: Without computing, is 3/4 ÷ 1/4 greater than or less than 3/4? Explain.
✓ Answer: Greater. The divisor 1/4 is less than 1, so more than one group of 1/4 fits in 3/4 (the quotient is 3).
Lesson 6-12
Least Common Multiple
List Multiples to Find the LCM · Use Prime Factors · Repeating-Schedule Problems
6.NS.B.4
Why this unlocks Lesson 6.12: This lesson builds on 4 skills from earlier math. Start with “Skip-count to list the multiples of a number”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Skip-count to list the multiples of a numberGr 4
2Recall multiplication facts through 12 × 12Gr 3
3Decide whether one number divides another evenlyGr 4
4Write a whole number as a product of prime factors6-7
2-Minute Quick Check
List the first 5 multiples of 6. 8 × 7? Does 4 divide 30 evenly?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: What is the 7th multiple of 8?
✓ Answer: 56
Task 2: Compute 9 × 12.
✓ Answer: 108
Task 3: Does 6 divide 45 evenly? Explain how you know.
✓ Answer: No. 6 × 7 = 42 and 6 × 8 = 48, so 45 ÷ 6 leaves a remainder of 3. 45 is not a multiple of 6.
Task 4: Write 24 as a product of prime factors.
✓ Answer: 24 = 2 × 2 × 2 × 3
Lesson 6-13
Prime Factorization
Prime and Composite Numbers · Factor Trees · Exponent Form of a Prime Factorization
6.NS.B.4
Why this unlocks Lesson 6.13: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication facts through 12 × 12 to split a number into a factor pair”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication facts through 12 × 12 to split a number into a factor pairGr 3
2Decide whether a number within 100 is prime or compositeGr 4
3Test divisibility by 2, 3 and 5 and divide out that factorGr 4
4Evaluate a power such as 2³ as repeated multiplication6-3
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 7 × 9.
✓ Answer: 63
Task 2: Is 51 prime or composite? Give a factor pair as evidence.
✓ Answer: Composite: 51 = 3 × 17.
Task 3: 135 ends in 5, so 5 is a factor. Compute 135 ÷ 5.
✓ Answer: 27
Task 4: Evaluate 2³ × 3.
✓ Answer: 24
Lesson 6-14
The Distributive Property
Expand a(b + c) · Factor Out the Greatest Common Factor · Distribute for Mental Math
6.EE.A.3
Why this unlocks Lesson 6.14: This lesson builds on 4 skills from earlier math. Start with “Multiply a one-digit number by a multiple of ten and add the partial products”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12.
Fluency Needed First · 4 Prerequisite Skills
1Multiply a one-digit number by a multiple of ten and add the partial productsGr 4
2Read a term such as 4n as 4 times n6-5
3Find the greatest common factor of two whole numbers6-7
4Split a rectangle into two parts and add their areasGr 3
2-Minute Quick Check
9 × 30 + 9 × 4? GCF of 12 and 18? What does 5y mean?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 8 × 40 + 8 × 3.
✓ Answer: 344
Task 2: What does 4n mean? Write it as a product and describe it in words.
✓ Answer: 4 × n: four groups of n.
Task 3: Find the greatest common factor of 24 and 36.
✓ Answer: 12
Task 4: A rectangle is 6 units tall. Its width is split into parts of 10 and 4 units. Add the two part areas to find the total area.
✓ Answer: 84
Lesson 6-15
Simplify Algebraic Expressions
Like Terms and Constants · Combine Like Terms · Distribute, Then Combine
6.EE.A.3
Why this unlocks Lesson 6.15: This lesson builds on 4 skills from earlier math. Start with “Name the coefficient, variable and constant in an expression”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents.
Fluency Needed First · 4 Prerequisite Skills
1Name the coefficient, variable and constant in an expression6-5
2Read a variable written alone, such as a, as 1a6-5
3Expand an expression with the distributive property6-14
4Substitute a value into an expression to check equivalence6-5
2-Minute Quick Check
Coefficient of y in y + 8? Expand 2(3a + 4). Evaluate 5x + 1 when x = 4.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Name the coefficient and the constant in 9k + 4.
✓ Answer: The coefficient is 9 and the constant is 4.
Task 2: In the term p, what is the coefficient?
✓ Answer: 1
Task 3: Expand 3(2w + 5).
✓ Answer: 6w + 15
Task 4: Evaluate 4x + 7 when x = 3.
✓ Answer: 19
Unit 79 lessons
Unit 7: Integers, Rational Numbers, and the Coordinate Plane
Place any number on a number line and read distance from it. Absolute value, comparison, and coordinate distance are all the same number-line move.
Lesson 7-1
Explore Integers and Their Opposites
Integers and Their Opposites · Integers in Real Situations
6.NS.C.5
Why this unlocks Lesson 7.1: This lesson builds on 4 skills from earlier math. Start with “Place whole numbers on a number line”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it.
Fluency Needed First · 4 Prerequisite Skills
1Place whole numbers on a number lineGr 4
2Count forward and backward through zeroGr 5
3Read a negative sign as “below zero” or “the opposite of”Gr 6
4Simplify an algebraic expression carried in from Unit 66-6
2-Minute Quick Check
What is the opposite of −6? Where does −3 sit compared with 0?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: On a number line marked in ones, what number is three steps right of 2?
✓ Answer: 5
Task 2: Start at 2 and move backward 5 steps. Where do you land?
✓ Answer: -3
Task 3: Write the integer for 7 degrees below zero.
✓ Answer: -7
Task 4: Simplify 3x + 4x.
✓ Answer: 7x
Lesson 7-2
Represent Rational Numbers and Their Opposites on the Number Line
Rational Numbers on a Number Line · Opposites of Rational Numbers
6.NS.C.6.c
Why this unlocks Lesson 7.2: This lesson builds on 4 skills from earlier math. Start with “Place fractions and decimals between whole numbers”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Move among a fraction, a decimal, and a percent.
Fluency Needed First · 4 Prerequisite Skills
1Place fractions and decimals between whole numbersGr 5
2Name the two consecutive integers a number falls betweenGr 5
3Convert between a fraction and a decimal4-2
4Partition a number-line interval into equal partsGr 4
2-Minute Quick Check
Which two integers does 2.7 fall between? Where does −11/2 go?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write the decimal halfway between 1 and 2.
✓ Answer: 1.5
Task 2: Between which two consecutive integers is -2.6?
✓ Answer: -3 and -2
Task 3: Write 3/5 as a decimal.
✓ Answer: 0.6
Task 4: An interval from 0 to 1 is split into 4 equal parts. What is the length of one part?
✓ Answer: 1/4
Lesson 7-3
Understand Absolute Value of Rational Numbers
Absolute Value as Distance · Absolute Value and Total Distance
6.NS.C.7.c
Why this unlocks Lesson 7.3: This lesson builds on 4 skills from earlier math. Start with “Count the distance between two points on a number line”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it.
Fluency Needed First · 4 Prerequisite Skills
1Count the distance between two points on a number lineGr 4
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: What is the distance from -4 to 0?
✓ Answer: 4
Task 2: What is the opposite of -7?
✓ Answer: 7
Task 3: Evaluate the absolute value of -9.
✓ Answer: 9
Task 4: A path goes 3 units to zero, then 5 more units. What total distance is traveled?
✓ Answer: 8
Lesson 7-4
Compare and Order Integers and Rational Numbers
Comparing Numbers on a Number Line · Ordering Integers and Rational Numbers
6.NS.C.7.a
Why this unlocks Lesson 7.4: This lesson builds on 4 skills from earlier math. Start with “Use < and > correctly”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it; Order a list of numbers least to greatest; Move among a fraction, a decimal, and a percent.
Fluency Needed First · 4 Prerequisite Skills
1Use < and > correctlyGr 3
2Place numbers on a number line7-2
3Compare decimals and fractions4-2
4Read left-is-less, right-is-greater on a number line7-1
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Fill the blank with < or >: -8 __ -3.
✓ Answer: <
Task 2: Order -1, 3, -4, 0 from least to greatest.
✓ Answer: -4, -1, 0, 3
Task 3: Which is greater: 3/4 or 0.7?
✓ Answer: 3/4
Task 4: Point A is at -5 and point B is at -2. Which point is farther right?
✓ Answer: B
Lesson 7-5
Represent Rational Numbers on the Coordinate Plane
Plotting Points on the Coordinate Plane · Reflecting Points Across the Axes
6.NS.C.6.b
Why this unlocks Lesson 7.5: This lesson builds on 4 skills from earlier math. Start with “Plot ordered pairs in Quadrant I”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Plot ordered pairs in Quadrant I3-4
2Read the sign of each coordinate to name a quadrantGr 6
3Move left or right first, then up or down from the originGr 5
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: From the origin, move 2 right and 6 up. Give the ordered pair.
✓ Answer: (2, 6)
Task 2: Which quadrant contains (-3, 4)?
✓ Answer: Quadrant II
Task 3: Describe the moves from the origin to (-4, -2).
✓ Answer: 4 left, 2 down
Task 4: An axis counts by 5. What value is two tick intervals left of zero?
✓ Answer: -10
Lesson 7-6
Determine Distance on the Coordinate Plane
Distance Between Points on a Grid · Comparing Distances on the Plane
6.NS.C.8
Why this unlocks Lesson 7.6: This lesson builds on 4 skills from earlier math. Start with “Subtract integers on a number line”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Subtract integers on a number line7-4
2Use absolute value to report a distance7-3
3Read the coordinates off a plotted point7-5
4Add two distances when a segment crosses an axis7-3
2-Minute Quick Check
Distance from (−5, 2) to (3, 2)? What is |−5 − 3|?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 6 − (-2).
✓ Answer: 8
Task 2: Evaluate the absolute value of -12.
✓ Answer: 12
Task 3: A plotted point is 3 units left and 5 units up from the origin. Give its coordinates.
✓ Answer: (-3, 5)
Task 4: Find the horizontal distance from (-4, 2) to (7, 2).
✓ Answer: 11
Lesson 7-7
Represent Polygons on the Coordinate Plane
Polygons and Perimeter on the Plane · Designing Polygons with a Scale
6.G.A.3
Why this unlocks Lesson 7.7: This lesson builds on 4 skills from earlier math. Start with “Plot several points and connect them in order”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane; Add a list of numbers to a running total; Find the area of a rectangle and of a triangle.
Fluency Needed First · 4 Prerequisite Skills
1Plot several points and connect them in order7-5
2Find horizontal and vertical distances between vertices7-6
3Add side lengths to find perimeterGr 4
4Find the area of a rectangle or triangle5-1, 5-2
2-Minute Quick Check
Plot (1,1), (5,1), (5,4), (1,4). What is the perimeter? The area?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Plot (0, 0), (4, 0), (4, 3), (0, 3) and connect in order. Name the shape.
✓ Answer: Rectangle
Task 2: Find the vertical distance from (2, -3) to (2, 5).
✓ Answer: 8
Task 3: Find the perimeter of a rectangle with side lengths 4 and 3.
✓ Answer: 14
Task 4: Find the area of a triangle with base 8 and height 5.
✓ Answer: 20
Lesson 7-8
Ordered Pairs in All Four Quadrants
Signs and Quadrants · Plotting and Moving Points · Side Lengths and Area on the Grid
6.NS.C.6.c
Why this unlocks Lesson 7.8: This lesson builds on 4 skills from earlier math. Start with “Plot and read a point in Quadrant I: across first, then up”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane; Place a number on a scaled number line and read distance from it.
Fluency Needed First · 4 Prerequisite Skills
1Plot and read a point in Quadrant I: across first, then upGr 5
2Locate integers on horizontal and vertical number lines7-2
3Find the distance between two points that share a coordinate7-6
4Find the perimeter and area of a rectangleGr 4
2-Minute Quick Check
What integer is 5 units left of 3? Describe how to reach (4, 6) from the origin. Area of a 7-by-2 rectangle?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Describe how to plot (5, 3) starting at the origin.
✓ Answer: Move 5 units right along the x-axis, then 3 units up.
Task 2: What integer is 6 units to the left of 2 on a number line?
✓ Answer: -4
Task 3: How far apart are the points (−3, 5) and (4, 5)?
✓ Answer: 7
Task 4: A rectangle is 6 units by 3 units. What is its perimeter?
✓ Answer: 18
Lesson 7-9
Reflect Points Across Axes
Reflect Across the x-Axis · Reflect Across the y-Axis · Distance Between a Point and Its Image
6.NS.C.6.b
Why this unlocks Lesson 7.9: This lesson builds on 4 skills from earlier math. Start with “Name the quadrant of a point from the signs of its coordinates”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Name the quadrant of a point from the signs of its coordinates7-8
2Find the opposite of an integer7-1
3Recognize points on an axis by a zero coordinate7-8
4Find a point's distance from an axis using absolute value7-3
2-Minute Quick Check
Opposite of −6? Quadrant of (4, −2)? Distance from (−5, 1) to the y-axis?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Which quadrant holds (−6, −1)? Explain using the signs.
✓ Answer: Quadrant III, because both coordinates are negative.
Task 2: What is the opposite of −9?
✓ Answer: 9
Task 3: Is the point (0, −5) on the x-axis or the y-axis? Explain.
✓ Answer: The y-axis, because its x-coordinate is 0.
Task 4: How far is the point (−7, 3) from the y-axis?
✓ Answer: 7
Unit 87 lessons
Unit 8: Equations and Inequalities
Inverse operations, and substituting a value back in to check. Students who do not know what undoes an operation cannot solve or verify anything in this unit.
Lesson 8-1
Understand Equations and Their Solutions
Equations and Solutions · Writing Equations
6.EE.B.5
Why this unlocks Lesson 8.1: This lesson builds on 4 skills from earlier math. Start with “Substitute a value into an expression and evaluate”, then the rest, so lesson time goes to the new idea. The core skill behind it: Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Substitute a value into an expression and evaluate6-5
2Read an equals sign as a balance, not as “the answer is”Gr 4
3Translate a sentence into an equation6-5
4Compare two sides to decide whether they match6-6
2-Minute Quick Check
Is x = 4 a solution of 3x = 12? Write an equation for “three times a number is 15.”
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Evaluate 3x + 2 when x = 4.
✓ Answer: 14
Task 2: Find the missing number: 7 + 5 = __ + 4.
✓ Answer: 8
Task 3: Write an equation for a number n plus 6 is 15.
✓ Answer: n + 6 = 15
Task 4: Does x = 5 make 2x + 1 = 11 true? Explain.
✓ Answer: Yes; 2 × 5 + 1 = 11
Lesson 8-2
Write and Solve Equations Using Addition or Subtraction
Addition Equations · Subtraction Equations
6.EE.B.7
Why this unlocks Lesson 8.2: This lesson builds on 4 skills from earlier math. Start with “Recall addition and subtraction facts and their inverses”, then the rest, so lesson time goes to the new idea. The core skill behind it: Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Recall addition and subtraction facts and their inversesGr 3
2Add and subtract multi-digit numbers, decimals, and fractionsGr 5
3Name the inverse of adding or subtracting a numberGr 4
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Find the missing number: __ + 7 = 19.
✓ Answer: 12
Task 2: Compute 5.2 − 1.85.
✓ Answer: 3.35
Task 3: Which operation undoes subtracting 8?
✓ Answer: Adding 8
Task 4: Check x = 9 in x + 6 = 15. State the left-side value.
✓ Answer: 15
Lesson 8-3
Write and Solve Equations Using Multiplication or Division
Multiplication Equations · Division Equations
6.EE.B.7
Why this unlocks Lesson 8.3: This lesson builds on 4 skills from earlier math. Start with “Recall multiplication and division facts”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Name the inverse of an operation and check by substituting back; Multiply and divide with fractions and mixed numbers.
Fluency Needed First · 4 Prerequisite Skills
1Recall multiplication and division factsGr 3
2Divide multi-digit numbers and decimals2-7
3Name the inverse of multiplying or dividing by a numberGr 4
4Divide by a fraction to undo multiplying by one6-2
2-Minute Quick Check
6x = 42, so x = ? x ÷ 5 = 9, so x = ? What undoes “× 4”?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 9 × 7.
✓ Answer: 63
Task 2: Compute 12.6 ÷ 3.
✓ Answer: 4.2
Task 3: Which operation undoes multiplying by 6?
✓ Answer: Dividing by 6
Task 4: Compute 12 ÷ 3/4.
✓ Answer: 16
Lesson 8-4
Write and Represent Inequalities
Writing Inequalities · Representing Inequalities
6.EE.B.8
Why this unlocks Lesson 8.4: This lesson builds on 4 skills from earlier math. Start with “Use <, >, ≤, and ≥ correctly”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Which symbol means greater than or equal to?
✓ Answer: ≥
Task 2: Write an inequality for a number n that is at least 12.
✓ Answer: n ≥ 12
Task 3: On a number line marked in ones, what number is two steps left of 1?
✓ Answer: -1
Task 4: Does 7 satisfy n < 5? Explain.
✓ Answer: No; 7 is greater than 5
Lesson 8-5
Understand Inequalities and Their Solutions
Graphing Inequalities · Testing Solutions
6.EE.B.5
Why this unlocks Lesson 8.5: This lesson builds on 4 skills from earlier math. Start with “Mark a boundary point on a number line and shade one direction”, then the rest, so lesson time goes to the new idea. The core skill behind it: Place a number on a scaled number line and read distance from it.
Fluency Needed First · 4 Prerequisite Skills
1Mark a boundary point on a number line and shade one direction8-4
2Choose an open or closed circle from the symbol8-4
3Substitute and test several candidate values8-1
4Compare a value against the boundary number7-4
2-Minute Quick Check
Graph x ≥ 3. Open circle or closed? Is 3 itself a solution?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Describe the graph of x > 3 on a number line.
✓ Answer: Open circle at 3; shade right
Task 2: For x ≤ 4, is the circle at 4 open or closed?
✓ Answer: Closed
Task 3: Which of 2, 5, 8 satisfy x ≥ 5?
✓ Answer: 5 and 8
Task 4: Which is greater: -2 or -6?
✓ Answer: -2
Lesson 8-6
Solve and Graph Inequalities
Solve One-Step Inequalities · Open and Closed Endpoints · Describe the Solution Set
6.EE.B.8
Why this unlocks Lesson 8.6: This lesson builds on 4 skills from earlier math. Start with “Solve a one-step equation with an inverse operation”, then the rest, so lesson time goes to the new idea. The core skill behind it: Name the inverse of an operation and check by substituting back; Place a number on a scaled number line and read distance from it.
Fluency Needed First · 4 Prerequisite Skills
1Solve a one-step equation with an inverse operation8-2
2Read the symbols <, >, ≤ and ≥ in words8-4
3Test whether a value makes an inequality true by substitution8-5
4Locate whole numbers on a scaled number lineGr 4
2-Minute Quick Check
Solve x + 5 = 13. Is 4 a solution of x ≤ 4? Read x > 9 in words.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Solve x + 7 = 19.
✓ Answer: 12
Task 2: Read x ≥ 4 in words.
✓ Answer: x is greater than or equal to 4.
Task 3: Is 6 a solution of x < 6? Explain.
✓ Answer: No. 6 < 6 is false, because 6 equals 6.
Task 4: A number line has tick marks every 5 units, starting at 0. What number sits 4 tick marks to the right of 0?
✓ Answer: 20
Lesson 8-7
Equations and Inequalities Problem Solving
Choose an Equation or an Inequality · Translate Constraint Words · Solve and Check in Context
6.EE.B.7
Why this unlocks Lesson 8.7: This lesson builds on 4 skills from earlier math. Start with “Translate at most, at least, more than and fewer than into inequality symbols”, then the rest, so lesson time goes to the new idea. The core skill behind it: Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Translate at most, at least, more than and fewer than into inequality symbols8-4
2Solve a one-step equation with addition or subtraction8-2
3Solve a one-step equation with multiplication or division8-3
4Check a solution by substituting it into the original equation8-1
2-Minute Quick Check
Solve m + 15 = 40. Solve 7k = 63. Write "fewer than 10" for b.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Write "at least 30" as an inequality for m minutes.
✓ Answer: m ≥ 30
Task 2: Solve p + 17 = 52.
✓ Answer: 35
Task 3: Solve 6w = 54.
✓ Answer: 9
Task 4: Is 9 a solution of 4x = 32? Show the check.
✓ Answer: No. 4 × 9 = 36, not 32. The solution is x = 8.
Unit 94 lessons
Unit 9: Relationships Between Two Variables
The unit rate from Unit 3, the table-to-graph move from Unit 3, and substitution from Units 6 and 8. Unit 9 asks students to carry one relationship through a table, a graph, and an equation.
Lesson 9-1
Explore Relationships Between Two Variables
Independent and Dependent Variables · Tables of Values
6.EE.C.9
Why this unlocks Lesson 9.1: This lesson builds on 4 skills from earlier math. Start with “Evaluate an expression for several input values”, then the rest, so lesson time goes to the new idea. The core skill behind it: Recall multiplication and division facts through 12 × 12; Scale a pair of numbers by the same factor; Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Evaluate an expression for several input values6-5
2Build a table of values from a rule3-3
3Multiply a unit rate by a quantity3-2
4Tell which quantity is chosen and which one respondsGr 6
2-Minute Quick Check
Songs cost $2 each. Which quantity is independent? Complete the table for 1, 2, and 3 songs.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Evaluate 4x + 1 when x = 3.
✓ Answer: 13
Task 2: For y = 2x + 1, give y for x = 0, 1, 2 in that order.
✓ Answer: 1, 3, 5
Task 3: Tickets cost $6 each. What is the cost in dollars of 7 tickets?
✓ Answer: 42
Task 4: You choose how many notebooks to buy. The total cost changes with that number. Which quantity is independent?
✓ Answer: Number of notebooks
Lesson 9-2
Analyze Graphs of Relationships Between Two Variables
From Table to Graph · Analyzing a Graph
6.EE.C.9
Why this unlocks Lesson 9.2: This lesson builds on 4 skills from earlier math. Start with “Plot ordered pairs taken from a table”, then the rest, so lesson time goes to the new idea. The core skill behind it: Plot and read ordered pairs on a coordinate plane.
Fluency Needed First · 4 Prerequisite Skills
1Plot ordered pairs taken from a table3-4, 7-5
2Read a point's coordinates off a graph7-5
3Read the scale on each axis3-4
4Extend a pattern to predict the next point3-3
2-Minute Quick Check
In (x, y), which value comes first? Read the point at 4 across and 12 up.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: A table has x = 3 and y = 12 in one row. Write its ordered pair.
✓ Answer: (3, 12)
Task 2: A graph point is 5 units right and 20 units up. Give its coordinates.
✓ Answer: (5, 20)
Task 3: A y-axis is labeled 0, 10, 20, 30. What value lies halfway between 10 and 20?
✓ Answer: 15
Task 4: Continue the points (1, 4), (2, 8), (3, 12) using y = 4x. Give the point for x = 4.
✓ Answer: (4, 16)
Lesson 9-3
Write Equations to Represent Relationships Between Two Variables
Equations from Tables · Equations from Situations
6.EE.C.9
Why this unlocks Lesson 9.3: This lesson builds on 4 skills from earlier math. Start with “Find the unit rate from a table”, then the rest, so lesson time goes to the new idea. The core skill behind it: Scale a pair of numbers by the same factor.
Fluency Needed First · 4 Prerequisite Skills
1Find the unit rate from a table3-2
2Divide y by x to find the constant factor3-2
3Write an expression using a variable6-5
4Test an equation against every row of the table8-1
2-Minute Quick Check
A proportional table gives (1, 3), (2, 6), (3, 9). Write y in terms of x.
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: A proportional table contains (2, 10), (4, 20), (6, 30). Find the unit rate y/x.
✓ Answer: 5
Task 2: For x = 8 and y = 28, compute y/x.
✓ Answer: 3.5
Task 3: Write an expression for the cost of n notebooks at $4 each.
✓ Answer: 4n
Task 4: Does y = 3x fit every pair (1, 3), (2, 6), (3, 10)? Explain.
✓ Answer: No; (3, 10) fails
Lesson 9-4
Apply Two-Variable Relationships to Solve Problems
Substitute and Solve · Working Backward to a Goal
6.EE.C.9
Why this unlocks Lesson 9.4: This lesson builds on 4 skills from earlier math. Start with “Substitute into an equation and evaluate”, then the rest, so lesson time goes to the new idea. The core skill behind it: Follow order of operations, including parentheses and exponents; Name the inverse of an operation and check by substituting back.
Fluency Needed First · 4 Prerequisite Skills
1Substitute into an equation and evaluate6-5
2Solve a one-step equation for the other variable8-2, 8-3
3Multiply and divide with decimals2-7
4Read the answer back in the situation's units3-2
2-Minute Quick Check
Muffins cost $3 each, so y = 3x. How many muffins can you buy with $24?