Reveal Math · McGraw Hill Alignment · Grade 6

Fluency & Readiness Index

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 Lessons 312 Prerequisite Skills 9 Units (Units 1–9) 624 Interactive Tasks
Unit 1 6 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
1 Multiply a one-digit number by a multiple of 10 Gr 4
2 Round a two-digit number to the nearest ten to estimate a product Gr 4
3 Decide when equal groups call for multiplication, not addition Gr 3
4 Decompose a decimal into place-value parts that rebuild it exactly Gr 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
1 Find a unit fraction of a whole number by dividing Gr 4
2 Multiply a fraction by a whole number Gr 5
3 Predict whether a fraction of a number is smaller or larger than the number Gr 5
4 Decompose a decimal into place-value parts that rebuild it exactly Gr 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
1 Multiply a decimal by 10 and by 100 Gr 5
2 Multiply a decimal by a whole number Gr 5
3 Divide a multiple of ten by a two-digit divisor Gr 5
4 Convert hours to minutes Gr 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
1 Multiply three whole numbers in any order Gr 4
2 Find the volume of a rectangular prism with V = l × w × h Gr 5
3 Multiply one box's volume by the number of identical boxes Gr 4
4 Add the volumes of two prisms to find a combined volume Gr 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
1 Multiply a decimal money value by 100 Gr 5
2 Extend a pattern by repeatedly applying an add rule Gr 4
3 Find a later term as start + number of steps × rule Gr 5
4 Recognize multiples of 5 by their last digit Gr 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
1 Multiply to find the total in equal groups Gr 3
2 Interpret “times as many” as multiplication, not addition Gr 4
3 Multiply a two-digit number by a one-digit number Gr 4
4 Divide to find the number of equal groups Gr 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 2 12 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
1 Order a list of up to 10 whole numbers least to greatest Gr 4
2 Read a scaled number line and say what one tick mark is worth Gr 4
3 Tally and count a set of up to 30 responses without losing place Gr 3
4 Compare two whole numbers and say which is greater Gr 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
1 Decide which interval a value belongs to (is 19 in 10–19 or 20–29?) Gr 5
2 Sort up to 20 values into given intervals and count each bin Gr 5
3 Add a column of one- and two-digit numbers to a total Gr 4
4 Read a graph scale that skip-counts by 2, 5, or 10 Gr 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
1 Order up to 12 numbers least to greatest Gr 4
2 Find the middle position of an odd-length list by counting in from both ends Gr 5
3 Find the number halfway between two numbers Gr 5
4 Divide a two-digit sum by 2 Gr 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
1 Order a data set and find its median 2-3
2 Split an ordered list into a lower half and an upper half 2-3
3 Subtract two-digit whole numbers Gr 4
4 Plot a point at the right place on a scaled number line Gr 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
1 Find the maximum and minimum of a data set Gr 4
2 Subtract whole numbers, including across zeros Gr 4
3 Locate Q1 and Q3 in an ordered set 2-4
4 Compare two differences to say which spread is larger Gr 4
2-Minute Quick Check
Range of 12, 7, 19, 4, 15? If Q1 = 6 and Q3 = 17, what is Q3 − Q1?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Give the minimum and maximum of 18, 7, 24, 11, 16.
✓ Answer: Minimum 7; maximum 24
Task 2: Compute 100 − 37.
✓ Answer: 63
Task 3: For 2, 4, 6, 8, 10, 12, 14, 16, find Q1 and Q3 using the medians of the two halves.
✓ Answer: Q1 = 5; Q3 = 13
Task 4: Set A ranges from 12 to 30. Set B ranges from 8 to 31. Which has the larger range?
✓ Answer: Set B
Lesson 2-6

Divide Multi-Digit Numbers Using an Algorithm

Estimate Quotients · Whole-Number Division · Interpret Remainders
6.NS.B.2

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
1 Recall multiplication and division facts through 12 × 12 Gr 3
2 Use place value to divide hundreds, tens, and ones Gr 4
3 Subtract multi-digit whole numbers with regrouping Gr 4
4 Estimate a quotient and check with multiplication Gr 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
1 Recall division facts through 12 × 12 Gr 3–4
2 Divide multi-digit whole numbers with the standard algorithm Gr 5
3 Multiply and divide by 10, 100, 1000 by shifting the decimal point Gr 5
4 Line up decimal places and annex zeros to keep dividing Gr 5
2-Minute Quick Check
84 ÷ 4? 4.8 ÷ 2? 0.6 × 100?
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
1 Add a list of 5–10 two-digit numbers accurately Gr 4
2 Divide a three-digit number by a one-digit number Gr 5
3 Count how many values are in a data set Gr 3
4 Find a missing addend that reaches a given total Gr 4
2-Minute Quick Check
Mean of 10, 14, 8, 12, 16? What must be added to 48 to reach 60?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Add 12 + 18 + 15 + 25 + 10.
✓ Answer: 80
Task 2: Compute 144 ÷ 6.
✓ Answer: 24
Task 3: How many values are in 4, 4, 7, 9, 9, 9?
✓ Answer: 6
Task 4: Find the missing addend: 18 + 22 + __ = 65.
✓ Answer: 25
Lesson 2-9

Describe Data by Mean Absolute Deviation

Finding the MAD · Comparing Sets with MAD
6.SP.B.5.c

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
1 Find the mean of a data set 2-8
2 Subtract to find the distance between two numbers and keep it positive Gr 5
3 Add a list of distances Gr 4
4 Divide a total by the number of values 2-8
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
1 Compute mean, median, range, IQR, and MAD on demand 2-3, 2-5, 2-8, 2-9
2 Spot an outlier in a short list 2-1, 2-2
3 Read a dot plot or histogram as symmetric or skewed 2-1, 2-2
4 Compare two measures and say which is pulled by an extreme value 2-8
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
1 Name the place value of each digit in a decimal through thousandths Gr 5
2 Annex zeros so two decimals have the same number of decimal places Gr 5
3 Add multi-digit whole numbers with regrouping Gr 4
4 Subtract multi-digit whole numbers by regrouping across zeros Gr 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
1 Multiply a multi-digit whole number by a two-digit whole number Gr 5
2 Recall multiplication facts through 12 × 12 Gr 3
3 Round a decimal to the nearest whole number to estimate Gr 5
4 Line up decimal points to add and subtract decimals 2-11
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 3 10 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
1 Recall multiplication facts through 12 × 12 Gr 3
2 Skip-count by any one-digit number Gr 3
3 Count each part of a part-whole picture, then add the parts for the total Gr 2–4
4 Write a fraction naming a part of a whole Gr 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
1 Divide a multi-digit number by a one- or two-digit number Gr 5
2 Divide to find the amount for one, and read “per” as “for each one” Gr 4
3 Keep a repeating pattern going in a table Gr 4
4 Attach the correct unit label to a computed number Gr 4
2-Minute Quick Check
60 ÷ 4? If 4 pencils cost $2, what does 1 pencil cost?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Compute 168 ÷ 12.
✓ Answer: 14
Task 2: Five notebooks cost $15. What is the cost in dollars for one notebook?
✓ Answer: 3
Task 3: A table gives 1 bag → 4 marbles, 2 bags → 8, 3 bags → 12. How many marbles for 5 bags?
✓ Answer: 20
Task 4: A cyclist rides 24 miles in 2 hours. State the unit rate with its unit.
✓ Answer: 12 miles per hour
Lesson 3-3

Equivalent Ratios Using Tables

Modeling Clay Recipe (ratio tables) · The Bridge Strategy (Scaling Ratios)
6.RP.A.3.a

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
1 Multiply both numbers of a pair by the same factor Gr 4
2 Recall multiplication facts through 12 × 12 Gr 3
3 Write equivalent fractions by multiplying numerator and denominator Gr 4
4 Fill the next row of a growing pattern table Gr 5
2-Minute Quick Check
2 × 5 and 3 × 5? Complete: 2 : 3 = 8 : ___
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
1 Plot a point in Quadrant I from an ordered pair Gr 5
2 Read the scale on each axis before plotting Gr 5
3 Count right from the origin, then up Gr 5
4 Build a ratio table to generate the pairs 3-3
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
1 Compare two whole numbers Gr 2
2 Scale both parts of a ratio to hit a target value 3-3
3 Find a common multiple of two numbers Gr 5
4 Divide to find a unit rate for a fair comparison 3-2
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
1 Recall multiplication and division facts Gr 3
2 Recall customary facts: 1 ft = 12 in, 1 lb = 16 oz, 1 gal = 4 qt Gr 4–5
3 Multiply and divide by 10, 100, 1000 for metric steps Gr 5
4 Decide whether a conversion calls for multiplying or dividing Gr 5
2-Minute Quick Check
12 × 4? 1 foot = ___ inches? 2.5 m = ___ cm?
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
1 Multiply a whole number by a decimal Gr 5
2 Round a decimal to the nearest tenth or hundredth Gr 5
3 Multiply by a conversion factor and keep the unit label 3-6
4 Chain two conversions in order without losing a step 3-6
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.

Fluency Needed First · 4 Prerequisite Skills
1 Divide a money amount by a whole number Gr 5
2 Compare two decimals to the hundredths place Gr 5
3 Multiply a decimal by a whole number Gr 5
4 Find a unit rate as an amount per one 3-2
2-Minute Quick Check
$4.50 ÷ 6? Which is less: 0.75 or 0.8? 7 × 0.35?
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
1 Recall multiplication and division facts through 12 × 12 Gr 3
2 Find the greatest common factor of two numbers within 100 Gr 4
3 Find a missing factor: 5 × ? = 20 Gr 3
4 Write a ratio in the order a:b and name what each term counts 3-1
2-Minute Quick Check
15 ÷ 5? GCF of 15 and 25? 4 × ? = 28?
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
1 Recall common customary relationships: 12 in = 1 ft, 3 ft = 1 yd, 16 oz = 1 lb, 4 qt = 1 gal Gr 4
2 Multiply a whole number or decimal by 10, 100, or 1,000 Gr 5
3 Divide with a remainder and interpret it in a measurement Gr 4
4 Use an equivalent ratio to find a missing value 3-9
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 4 5 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
1 Read and write a fraction with denominator 100 Gr 4
2 Count shaded squares on a 100-square grid Gr 4
3 Write hundredths as a decimal Gr 5
4 Simplify a fraction over 100 Gr 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
1 Multiply and divide by 100 by shifting the decimal point Gr 5
2 Write a fraction as a decimal by dividing Gr 5
3 Recall benchmark equivalents for 1/2, 1/4, 1/5, 3/4, 1/10 Gr 4–5
4 Compare decimals to the hundredths place Gr 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
1 Find 10% of a number by shifting the decimal one place left Gr 5
2 Find 1% of a number by shifting two places left Gr 5
3 Halve and double a number mentally Gr 4
4 Round to a compatible number that is easy to take a percent of Gr 5
2-Minute Quick Check
10% of 320? 1% of 320? Half of 32?
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
1 Write a percent as a decimal 4-2
2 Multiply a whole number by a decimal Gr 5
3 Multiply a whole number by a fraction Gr 5
4 Compare two products to decide which is larger Gr 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
1 Divide a whole number by a decimal 2-7
2 Tell which number is the part and which is the whole 4-4
3 Divide by a benchmark fraction such as 1/4 or 1/2 Gr 5
4 Check an answer by multiplying back Gr 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 5 10 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
1 Multiply two- and three-digit numbers Gr 5
2 Find the area of a rectangle by multiplying length by width Gr 3–4
3 Tell a base from a slanted side and identify the perpendicular height Gr 5
4 Label an area answer in square units Gr 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
1 Multiply, then halve — or halve, then multiply Gr 5
2 Multiply a whole number by 1/2 Gr 5
3 Find the area of a rectangle or parallelogram 5-1
4 Identify the height drawn perpendicular to the chosen base 5-1
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
1 Add the two numbers inside parentheses before multiplying Gr 5
2 Multiply by 1/2 5-2
3 Follow order of operations inside a formula Gr 5
4 Split a figure into a rectangle plus triangles 5-2
2-Minute Quick Check
1/2 × (6 + 10) × 4 = ? Do the parentheses first.
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
1 Find the area of rectangles, triangles, and trapezoids on demand 5-1, 5-2, 5-3
2 Add and subtract areas Gr 4
3 Find a missing side length from the other labeled sides Gr 5
4 Multiply one part's area by the number of identical parts Gr 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
1 Multiply three factors in any order Gr 5
2 Multiply fractions and mixed numbers Gr 5
3 Find the area of a rectangular base 5-1
4 Label a volume answer in cubic units Gr 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
1 Name the faces, edges, and vertices of a prism and a pyramid Gr 5
2 Count the faces of a solid Gr 5
3 Find the area of each rectangle and triangle inside a net 5-1, 5-2
4 Match congruent faces in pairs Gr 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
1 Add a list of five or six areas Gr 4
2 Find the area of rectangles and triangles quickly 5-1, 5-2
3 Read a net and pair up matching faces 5-6
4 Double one face's area to cover a matching pair Gr 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
1 Find the area of a triangle 5-2
2 Find the area of a square or rectangular base 5-1
3 Multiply one triangular face's area by the number of identical faces Gr 4
4 Tell slant height from vertical height on a net or drawing 5-6
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
1 Find the area of a triangle with A = ½ × base × height 5-2
2 Multiply a decimal by a whole number Gr 5
3 Name how many sides a pentagon, hexagon, and octagon have Gr 3
4 Find half of a whole number or a decimal Gr 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
1 Rewrite a mixed number as an improper fraction Gr 5
2 Multiply a fraction by a whole number Gr 5
3 Multiply a decimal by whole numbers Gr 5
4 Find a missing factor by dividing the product by the known factors 5-5
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 6 15 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
1 Divide decimals and whole numbers 2-7
2 Name the reciprocal of a whole number and of a unit fraction Gr 5
3 Multiply a whole number by a fraction Gr 5
4 Read 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
1 Rewrite a mixed number as an improper fraction Gr 5
2 Multiply two fractions Gr 5
3 Find the reciprocal of any fraction 6-1
4 Simplify a fraction to lowest terms Gr 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
1 Multiply the same factor repeatedly Gr 4
2 Recall 2², 3², 4², 5², 2³, and 3³ Gr 5–6
3 Multiply by 10 repeatedly to build powers of 10 Gr 5
4 Count how many factors are written Gr 3
2-Minute Quick Check
6²? Write 2 × 2 × 2 × 2 as a power. 10³?
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
1 Evaluate a power 6-3
2 Follow order of operations with parentheses and exponents Gr 5
3 Read a fraction bar as a grouping symbol Gr 5
4 Translate “sum,” “product,” and “less than” into symbols Gr 5
2-Minute Quick Check
4 + 6 ÷ 2? 3 + 2² × 5? Which operation comes first?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: Evaluate 4³.
✓ Answer: 64
Task 2: Evaluate 2 + 3² × 4.
✓ Answer: 38
Task 3: Evaluate (8 + 4)/3.
✓ Answer: 4
Task 4: Write an expression for 7 less than the product of 3 and 5.
✓ Answer: 3 × 5 − 7
Lesson 6-5

Write and Evaluate Algebraic Expressions

Writing Algebraic Expressions · Evaluating Algebraic Expressions
6.EE.A.2

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
1 Substitute a number for a letter, then compute Gr 6
2 Follow order of operations after substituting 6-4
3 Read 4n as 4 × n Gr 6
4 Translate a word phrase into an expression 6-4
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
1 Combine like terms such as 5x + 2x Gr 6
2 Evaluate an expression at a chosen value 6-5
3 Apply the distributive property with numbers, e.g. 3(4 + 5) Gr 5
4 Compare two results to decide whether expressions match Gr 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
1 Recall multiplication and division facts Gr 3
2 List every factor pair of a number up to 100 Gr 4
3 List the first several multiples of a number Gr 4
4 Test divisibility by 2, 3, 5, 9, and 10 Gr 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
1 Find the GCF of two numbers 6-7
2 Apply the distributive property forward and backward 6-6
3 Combine like terms 6-6
4 Multiply a number by a sum Gr 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
1 Count how many unit fractions make one whole Gr 5
2 Multiply a whole number by a fraction Gr 5
3 Write the reciprocal of a fraction or whole number 6-1
4 Interpret a leftover part of a group in a word problem Gr 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
1 Rewrite a mixed number as an improper fraction Gr 5
2 Write the reciprocal of a fraction 6-9
3 Multiply two fractions Gr 5
4 Rewrite an improper fraction as a whole or mixed number Gr 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
1 Name the total (dividend) and the group size (divisor) in a story 6-9
2 Divide a fraction by a fraction using the reciprocal 6-1
3 Multiply a whole number by a fraction to check a quotient Gr 5
4 Compare a divisor to 1 to predict whether a quotient is greater or less than the dividend Gr 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
1 Skip-count to list the multiples of a number Gr 4
2 Recall multiplication facts through 12 × 12 Gr 3
3 Decide whether one number divides another evenly Gr 4
4 Write a whole number as a product of prime factors 6-7
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
1 Recall multiplication facts through 12 × 12 to split a number into a factor pair Gr 3
2 Decide whether a number within 100 is prime or composite Gr 4
3 Test divisibility by 2, 3 and 5 and divide out that factor Gr 4
4 Evaluate a power such as 2³ as repeated multiplication 6-3
2-Minute Quick Check
Is 39 prime or composite? 6 × 7? Evaluate 2³.
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
1 Multiply a one-digit number by a multiple of ten and add the partial products Gr 4
2 Read a term such as 4n as 4 times n 6-5
3 Find the greatest common factor of two whole numbers 6-7
4 Split a rectangle into two parts and add their areas Gr 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
1 Name the coefficient, variable and constant in an expression 6-5
2 Read a variable written alone, such as a, as 1a 6-5
3 Expand an expression with the distributive property 6-14
4 Substitute a value into an expression to check equivalence 6-5
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 7 9 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
1 Place whole numbers on a number line Gr 4
2 Count forward and backward through zero Gr 5
3 Read a negative sign as “below zero” or “the opposite of” Gr 6
4 Simplify an algebraic expression carried in from Unit 6 6-6
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
1 Place fractions and decimals between whole numbers Gr 5
2 Name the two consecutive integers a number falls between Gr 5
3 Convert between a fraction and a decimal 4-2
4 Partition a number-line interval into equal parts Gr 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
1 Count the distance between two points on a number line Gr 4
2 Name the opposite of a number 7-1
3 Report a distance as a positive number Gr 6
4 Add two distances to find a total Gr 3
2-Minute Quick Check
|−9|? How far is −4 from 0? How far from −4 to 3?
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
1 Use < and > correctly Gr 3
2 Place numbers on a number line 7-2
3 Compare decimals and fractions 4-2
4 Read left-is-less, right-is-greater on a number line 7-1
2-Minute Quick Check
−6 ___ −2. Order these: −3, 1.5, −1/2, 0.
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
1 Plot ordered pairs in Quadrant I 3-4
2 Read the sign of each coordinate to name a quadrant Gr 6
3 Move left or right first, then up or down from the origin Gr 5
4 Read a scaled axis before plotting 3-4
2-Minute Quick Check
Which quadrant holds (−4, 5)? Plot (3, −2).
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
1 Subtract integers on a number line 7-4
2 Use absolute value to report a distance 7-3
3 Read the coordinates off a plotted point 7-5
4 Add two distances when a segment crosses an axis 7-3
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
1 Plot several points and connect them in order 7-5
2 Find horizontal and vertical distances between vertices 7-6
3 Add side lengths to find perimeter Gr 4
4 Find the area of a rectangle or triangle 5-1, 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
1 Plot and read a point in Quadrant I: across first, then up Gr 5
2 Locate integers on horizontal and vertical number lines 7-2
3 Find the distance between two points that share a coordinate 7-6
4 Find the perimeter and area of a rectangle Gr 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
1 Name the quadrant of a point from the signs of its coordinates 7-8
2 Find the opposite of an integer 7-1
3 Recognize points on an axis by a zero coordinate 7-8
4 Find a point's distance from an axis using absolute value 7-3
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 8 7 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
1 Substitute a value into an expression and evaluate 6-5
2 Read an equals sign as a balance, not as “the answer is” Gr 4
3 Translate a sentence into an equation 6-5
4 Compare two sides to decide whether they match 6-6
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
1 Recall addition and subtraction facts and their inverses Gr 3
2 Add and subtract multi-digit numbers, decimals, and fractions Gr 5
3 Name the inverse of adding or subtracting a number Gr 4
4 Substitute the answer back in to check 8-1
2-Minute Quick Check
m − 11 = 25, so m = ? What undoes “+ 8”?
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
1 Recall multiplication and division facts Gr 3
2 Divide multi-digit numbers and decimals 2-7
3 Name the inverse of multiplying or dividing by a number Gr 4
4 Divide by a fraction to undo multiplying by one 6-2
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.

Fluency Needed First · 4 Prerequisite Skills
1 Use <, >, ≤, and ≥ correctly 7-4
2 Translate “at least,” “at most,” “no more than,” “more than” Gr 6
3 Place a number on a number line 7-2
4 Decide whether a given value makes a comparison true 8-1
2-Minute Quick Check
Write an inequality for “you must be at least 48 inches tall.” Is 50 a solution?
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
1 Mark a boundary point on a number line and shade one direction 8-4
2 Choose an open or closed circle from the symbol 8-4
3 Substitute and test several candidate values 8-1
4 Compare a value against the boundary number 7-4
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
1 Solve a one-step equation with an inverse operation 8-2
2 Read the symbols <, >, ≤ and ≥ in words 8-4
3 Test whether a value makes an inequality true by substitution 8-5
4 Locate whole numbers on a scaled number line Gr 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
1 Translate at most, at least, more than and fewer than into inequality symbols 8-4
2 Solve a one-step equation with addition or subtraction 8-2
3 Solve a one-step equation with multiplication or division 8-3
4 Check a solution by substituting it into the original equation 8-1
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 9 4 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
1 Evaluate an expression for several input values 6-5
2 Build a table of values from a rule 3-3
3 Multiply a unit rate by a quantity 3-2
4 Tell which quantity is chosen and which one responds Gr 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
1 Plot ordered pairs taken from a table 3-4, 7-5
2 Read a point's coordinates off a graph 7-5
3 Read the scale on each axis 3-4
4 Extend a pattern to predict the next point 3-3
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
1 Find the unit rate from a table 3-2
2 Divide y by x to find the constant factor 3-2
3 Write an expression using a variable 6-5
4 Test an equation against every row of the table 8-1
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
1 Substitute into an equation and evaluate 6-5
2 Solve a one-step equation for the other variable 8-2, 8-3
3 Multiply and divide with decimals 2-7
4 Read the answer back in the situation's units 3-2
2-Minute Quick Check
Muffins cost $3 each, so y = 3x. How many muffins can you buy with $24?
4 Worked Prerequisite Practice Tasks (Foundation Review)
Task 1: For C = 2.5n, find C when n = 6.
✓ Answer: 15
Task 2: Solve 4x = 28.
✓ Answer: 7
Task 3: Compute 18.9 ÷ 3.
✓ Answer: 6.3
Task 4: A runner travels at 8 miles per hour for 3 hours. State the distance with its unit.
✓ Answer: 24 miles