Equivalent Essentials Plan Equivalent Essentials
Lesson 1: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will use number lines to identify and generate equivalent fractions, focusing on the relationship between the number of parts and the size of each part.
Standards
5.NF.A.1, ALN HLC 1
Materials
Fraction Strips (or paper strips)
Individual Whiteboards
Colored Markers
Number Line Template
Key Vocabulary
Equivalent
Denominator (unit)
Numerator (count)
Interval
IM Routine
Notice and Wonder: Two different number lines showing the same point.
Instructional Steps
1
Warm-up: Notice & Wonder (5 min)
Show two number lines from 0 to 1. One is partitioned into halves, the other into fourths. Ask: "What do you notice? What do you wonder?" Highlight that 1/2 and 2/4 land on the same physical spot.
2
Launch: Folding the Frontier (10 min)
Give students a paper strip. Have them fold it in half and mark "1/2". Now, ask them to fold it again. "What happened to our halves? How many pieces now?" Relate the folding to the number line. When we double the number of pieces (denominator), we double the number of shaded pieces (numerator) to keep the same amount.
3
Activity: The Jump Map (15 min)
Use the slide deck practice. Practice "partitioning" a number line into thirds, then using a different color to partition each third into two smaller pieces. "What are these new pieces called?" (sixths). Track how many 1/6 jumps equal one 1/3 jump.
4
Synthesis & Cool Down (5 min)
Ask: "How can we tell if two fractions are equivalent just by looking at a number line?" (They share the same point). Exit Ticket: Find one equivalent fraction for 3/4 using a number line.
Teacher Tips
Wait Time: Give students time to count the intervals on the number line. Many students count the lines instead of the spaces.Visual Support: Use different colored markers for different denominators on the same number line.
Differentiation
For Support: Stick to doubling (halves to fourths, fourths to eighths) before moving to tripling.
For Extension: Challenge students to find a fraction equivalent to 2/3 with a denominator of 12.
Equivalent Essentials Slides Fraction Frontiers
Equivalent Essentials
Building a foundation for adding and subtracting fractions with unlike denominators.
Expedition 1: Equivalent Fractions
Notice and Wonder
0
1/2
1
0
1/4
2/4
3/4
1
What do you notice?
Look at the positions of 1/2 and 2/4.
What do you wonder?
Ask a question about these number lines.
The Secret of "Scaling"
When we double the number of parts in our whole...
We must also double the number of parts we count!
1
2
× 2 = × 2
2
4
"The denominator defines the size, the numerator defines the count."
Guided Practice
1. Partition this number line into 3 equal parts (thirds).
0
1
2. Now split each third in half. How many parts do we have now?
Check: Is 1/3 the same as 2/6?
Equivalent Essentials Activity Equivalent Expedition
Lesson 1 Activity: Finding Same-Size Fractions
Student Name
1. Spot the Match
Look at the two number lines below. Label the points and explain what you see.
0
1/2
1
0
____
1
My Observation:
2. The Split Method
First, partition the line into 4 equal parts (fourths). Then, use a dashed line to split each part into 2 smaller parts.
0
1
How many total parts are there now?
What is the new denominator?
3. Equivalence Check
Use your work from above. Fill in the missing numbers to show equivalent fractions.
1
4
=
___
___
3
4
=
___
___
5.NF.A.1 INTERVENTION FRONTIER EXPEDITION 01
Unlikely Sums Plan Unlikely Sums
Lesson 2: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will add fractions with unlike denominators by using area models to find a common partition (common denominator).
Standards
5.NF.A.1, ALN HLC 2
Materials
Grid Paper
Transparencies (optional)
Colored Pencils
Area Model Templates
Key Vocabulary
Common Denominator
Partition
Sum
Unit Fraction
IM Routine
How many do you see? (Grid with shaded squares representing fractions).
Instructional Steps
1
Warm-up: The Mixed Tray (5 min)
Show a rectangle split into halves vertically and thirds horizontally. "Can we count these pieces easily? Are they the same size?" Highlight the need for equal-sized pieces to add.
2
Launch: The Super-Grid (10 min)
Show 1/2 + 1/3. Draw two identical rectangles. Partition one vertically (halves) and one horizontally (thirds). Overlap them (virtually or with transparencies). "Look! Now both rectangles have 6 small squares. We found a way to make them the same size!"
3
Activity: Common Ground Models (15 min)
Students use the activity sheet to draw area models. For each problem (e.g., 1/4 + 1/2), they must create a common grid. Stress the ALN idea: "Find the smallest number of pieces that both fractions can be broken into."
4
Synthesis & Cool Down (5 min)
Ask: "Why do we need a common denominator to add?" (So we are counting pieces of the same size). Exit Ticket: Find a common denominator for 1/2 and 1/5 and draw the grid.
Teacher Tips
Vertical vs Horizontal: Always partition the first fraction with vertical lines and the second with horizontal lines. This makes the common grid visually obvious.Counting Squares: Have students physically count the small squares in the final grid to find the denominator.
Misconception Alert
Students often add the denominators (e.g., 1/2 + 1/3 = 2/5). The area model prevents this by showing that the "size" of the whole stays the same, only the number of pieces changes.
Unlikely Sums Slides Unlikely Sums
Adding with Unlike Denominators
1
2
1
3
=
?
?
The "Same Size" Rule
We can only add things that are the same size.
"You can't add 2 apples and 3 oranges and say you have 5 apples!"
1/2
1/3
The pieces are different sizes!
The Solution: The Common Grid
1/2 becomes 3/6
1/3 becomes 2/6
3/6 + 2/6 = 5/6
Guided Practice: 1/4 + 1/2
Step 1: Draw the grids
Draw one box for 1/4 (vertical) and one for 1/2 (horizontal).
Step 2: Partition both
Make sure both grids have the same total number of squares.
Use your whiteboard!
Unlikely Sums Activity Grid Masters
Finding Common Denominators with Area Models
Student Name
Strategy: To add fractions with different denominators, we need to partition them so the pieces are the same size.
1. Calculate: \(\frac{1}{2} + \frac{1}{4}\)
Draw \(\frac{1}{2}\) (Vertical)
Draw \(\frac{1}{4}\) (Horizontal)
The Common Grid (Overlap)
____
8
____
8
=
____
8
2. Calculate: \(\frac{1}{3} + \frac{2}{5}\)
Vertical Thirds
Horizontal Fifths
Total pieces?
Write your final equation below:
5.NF.A.1 Tier 2 Intervention
Common Grid Strategy
Difference Dynamics Plan Difference Dynamics
Lesson 3: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will subtract fractions with unlike denominators by finding the difference (distance) on a number line using common intervals.
Standards
5.NF.A.1, ALN HLC 2
Materials
Dry-erase markers (2 colors)
Number Line Workmats
Fraction comparison cards
Key Vocabulary
Difference
Distance
Partitioning
Common Interval
IM Routine
Which one doesn't belong? (Four subtraction expressions, one with like denominators).
Instructional Steps
1
Warm-up: The Tape Measure (5 min)
Ask: "What is the difference between 10 inches and 6 inches?" How did you find it? Connect to the number line. Subtraction is finding the distance between two points.
2
Launch: Matching the Marks (10 min)
Show 3/4 - 1/2. On a number line, mark 3/4 and 1/2. Ask: "Can we count the jumps from 1/2 to 3/4?" (No, the marks are different). "How can we make the marks match?" (Partition the 1/2 into 2/4). Now the distance is easy to see: 1 fourth.
3
Activity: Distance Detectives (15 min)
Students work on problems like 2/3 - 1/6. They must first plot both fractions, then re-partition the line so both use the same denominator, then count the distance.
4
Synthesis & Cool Down (5 min)
Discuss: "Why is it easier to find the difference when the marks are the same?" (We have a consistent unit to count). Exit Ticket: Solve 5/6 - 1/3 using a number line.
Strategic Choice
While Lesson 2 used area models, Lesson 3 uses number lines. This provides students with multiple entry points (HLC strategy). Students who struggle with area can see the "distance" more clearly on a linear model.
Scaffolding
Provide number lines that are already partially partitioned for students who struggle with physical drawing. Use "Benchmark" points (0, 1/2, 1) to help them orient their numbers.
Difference Dynamics Slides Fraction Frontiers
Difference Dynamics
Subtracting with Number Lines
What is "Difference"?
Difference = Distance
0
1
Subtraction is just finding the distance between two fractions on a number line.
The "Matching" Problem
3
4
−
1
2
It's hard to count the distance because the intervals (marks) don't match!
Strategy:
Change the marks so both fractions use the same unit.
1/2
2/4
How to Subtract
1
Partition
Find a denominator that works for both numbers.
2
Plot
Mark both fractions on the number line.
3
Count
Count the distance (jumps) between the two dots.
Ready? Let's try 5/6 − 1/3 on your boards!
Difference Dynamics Activity Distance Detectives
Lesson 3 Activity: Subtraction on the Number Line
Student Name
1. Calculate: \(\frac{2}{3} - \frac{1}{6}\)
Step A: Plot both points (\(\frac{2}{3}\) and \(\frac{1}{6}\))
0
1
Step B: Re-partition the line into sixths and find the distance
0
1
New Equation
____
6
-
1
6
=
____
6
2. Calculate: \(\frac{3}{4} - \frac{1}{8}\)
0
1
I partitioned the line into eighths.
I counted ____ jumps of distance.
Final Answer: ________
5.NF.A.1 DIFFERENCE MODEL EXPEDITION 03
Estimate Edge Plan Estimate Edge
Lesson 4: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will use benchmark fractions (0, 1/2, 1) to estimate the sum or difference of fractions with unlike denominators and assess the reasonableness of their answers.
Standards
5.NF.A.2, ALN HLC 2
Materials
Benchmark Number Lines
Estimation Sort Cards
Student "Reasonableness" Checklist
Key Vocabulary
Benchmark
Estimate
Reasonable
Approximate
IM Routine
Estimation Exploration: "Is 4/9 + 1/2 greater than or less than 1?"
Instructional Steps
1
Warm-up: Closer To? (5 min)
Show various fractions (e.g., 1/10, 5/11, 8/9). Students decide if each is closer to 0, 1/2, or 1. Discuss why. (e.g., "5/11 is just a bit less than half because half of 11 is 5.5").
2
Launch: The Balance Test (10 min)
Present a word problem: "Jada ran 7/8 of a mile. Diego ran 1/10 of a mile. About how far did they run together?" Use benchmark "zones" to estimate. 7/8 is almost 1. 1/10 is almost 0. So 1 + 0 = about 1.
3
Activity: Benchmark Battle (15 min)
Students use the activity sheet to estimate sums/differences *before* calculating. They check their calculated answers against their estimates to see if they are "reasonable." If they get 8/18 for 1/2 + 1/3, they should notice it's less than 1/2 and flag it as unreasonable.
4
Synthesis & Cool Down (5 min)
Ask: "How can estimating save us from making big mistakes?" Exit Ticket: Estimate if 11/12 - 4/9 is closer to 0, 1/2, or 1.
Common Error
Students often add numerators and denominators (1/2 + 1/3 = 2/5). Estimation is the primary tool to catch this. 2/5 is less than 1/2, so it cannot be the sum of 1/2 and something else.
HLC Connection
ALN HLC 2 emphasizes using benchmark fractions to build number sense. This lesson transitions from model-heavy calculation to mental estimation.
Estimate Edge Slides ?
~
≈
≈
?
~
~
≈
?
Estimation Skills
Estimate Edge
Using Benchmarks to Predict the Answer
The Three "Zones"
0
1/2
1
Small Zone
1/10, 2/12, 1/8
Numerator is tiny compared to denominator.
Middle Zone
4/9, 6/11, 5/12
Numerator is about half of denominator.
Full Zone
9/10, 11/12, 7/8
Numerator is almost the same as denominator.
Prediction Power
!
11/12 + 1/10
≈ 1 + ≈ 0 = ≈ 1
"If your calculated answer isn't near 1, something went wrong!"
Reasonableness Check
Problem:
1/2 + 1/3
My estimate is about 1 (or between 1/2 and 1).
5/6
Reasonable! It's close to 1.
2/5
Not Reasonable! 2/5 is less than 1/2.
Estimate Edge Activity The Reasonableness Check
Lesson 4: Estimation & Benchmarks
Explorer Name
1. Find Your Zone
Write each fraction in the box it is closest to.
4/9
1/12
11/13
5/11
2/15
7/8
9/10
6/14
CLOSEST TO 0
CLOSEST TO 1/2
CLOSEST TO 1
2. Estimate, then Solve
3/8 + 4/9
Estimate ≈
Exact Answer
11/12 - 1/8
Estimate ≈
Exact Answer
Fraction Detective
Someone calculated 5/6 - 1/2 and got an answer of 4/4. Use estimation to explain why that answer cannot be correct.
5.NF.A.2 Intervention Confidence: 0 - 1/2 - 1
Sharing Solutions Plan Sharing Solutions
Lesson 5: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will interpret a fraction as division of the numerator by the denominator (\(a/b = a \div b\)) using sharing contexts and tape diagrams.
Standards
5.NF.B.3, ALN HLC 3
Materials
Paper rectangles (to represent "sandwiches")
Tape Diagram Templates
Sharing Scenario Cards
Key Vocabulary
Dividend
Divisor
Quotient
Sharing
IM Routine
Which One Doesn't Belong? \(4 \div 5\), \(\frac{4}{5}\), \(5 \div 4\), 4 pieces shared by 5 people.
Instructional Steps
1
Warm-up: Equal Sharing (5 min)
Ask: "If 4 friends share 8 cookies, how many cookies does each person get?" (2). "What if 4 friends share 1 cookie? How many cookies does each person get?" (1/4). Introduce the idea that the fraction bar is actually a division symbol.
2
Launch: The 3-Sub Problem (10 min)
Scenario: 3 sandwiches shared by 4 people. Give students 3 paper rectangles. "How can we give everyone the same amount?" Model cutting each sandwich into 4 pieces. Everyone gets one piece from each sandwich. \(1/4 + 1/4 + 1/4 = 3/4\).
3
Activity: The Sharing Station (15 min)
Students use the activity sheet to represent sharing scenarios using tape diagrams. For each one, they write the division expression and the resulting fraction. Focus on the relationship: "Number of things being shared" goes on top, "Number of people/groups" goes on bottom.
4
Synthesis & Cool Down (5 min)
Discuss: "Why does \(3 \div 4 = 3/4\)?" (Because we are breaking 3 wholes into 4 equal parts). Exit Ticket: Write \(5 \div 8\) as a fraction and draw a quick tape diagram.
Visual Model
The tape diagram should show the total "length" (the dividend) being cut into equal segments (the divisor). Each segment is the quotient.
Misconception
Students often put the larger number as the numerator regardless of the context. Emphasize identifying "What is being shared?" as the numerator.
Sharing Solutions Slides Sharing Solutions
The Secret Connection Between Fractions and Division
3 ÷ 4
3
4
The Sandwich Problem
If 3 sandwiches are shared by 4 friends...
How much does each person get?
1/4
1/4
1/4
The Sharing Rule
What is shared?
Dividend
Numerator
BECOMES
Shared by how many?
Divisor
Denominator
Sharing Practice
Scenario A:
8 friends share 5 pizzas.
?
Write the fraction!
Scenario B:
4 boxes of clay are shared by 10 students.
?
Write the fraction!
Sharing Solutions Activity Sharing Station
Fractions as Division: \(a \div b = \frac{a}{b}\)
Station Worker
1. Model the Sharing
The Story:
5 friends share 3 large granola bars equally.
Division
3 ÷ 5
Fraction
3/5
Tape Diagram Model:
Whole 1
Whole 2
Whole 3
"Each person gets 1/5 of each whole. Since there are 3 wholes, they get 3/5 in total."
Your Turn:
4 kids share 2 bottles of juice equally.
Division Expression:
Final Fraction:
Draw your tape diagrams here:
2. Station Dash
\(7 \div 8\)
_____
\(2 \div 3\)
_____
\(1 \div 6\)
_____
5.NF.B.3 INTERVENTION THE SHARING RULE: IN ÷ OUT = IN/OUT
Scaling Shapes Plan Scaling Shapes
Lesson 6: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will find a fraction of a whole number (\(1/3 \times 12\) or \(2/3 \times 12\)) by using tape diagrams to show partitioning and scaling.
Standards
5.NF.B.4.a, ALN HLC 3
Materials
Counters or unifix cubes
Pre-drawn Tape Diagrams
"Fraction of" Task Cards
Key Vocabulary
Scaling
Unit Fraction
Product
Multiple
IM Routine
True or False: \(1/4\) of 12 is greater than \(1/2\) of 4.
Instructional Steps
1
Warm-up: Grouping (5 min)
Show 12 counters. "If I want to put these into 3 equal groups, how many are in each group?" (4). Connect this to "What is 1/3 of 12?" Introduce the "of" means multiplication in this context.
2
Launch: The Two-Step (10 min)
Model \(2/3\) of 12. Step 1: Find 1/3 (partition 12 into 3 groups = 4). Step 2: Scale it up (take 2 of those groups = 8). Use a tape diagram: Draw a tape representing 12, cut into 3 boxes, put a '4' in each box. Shade 2 boxes.
3
Activity: Scale it Down (15 min)
Students work through the activity sheet using tape diagrams for problems like \(3/4\) of 20 and \(2/5\) of 25. Emphasize: The denominator tells you how many boxes to draw. The whole number tells you what the entire tape is worth. The numerator tells you how many boxes to shade.
4
Synthesis & Cool Down (5 min)
Ask: "Is \(2/3\) of 12 the same as \(12 \times 2/3\)? Is it the same as \((12 \div 3) \times 2\)?". Exit Ticket: Solve \(3/5 \times 15\) using a quick diagram.
HLC 3 Context
This lesson bridges whole number division and fraction multiplication. It is the "conceptual anchor" for standard multiplication algorithms.
Mental Math Tip
Encourage students to say "divided by the bottom, times the top" once they have mastered the visual tape diagram.
Scaling Shapes Slides Expedition 06
Scaling Shapes
Taking a fraction of a whole.
The Two-Step Secret
2/3 of 12
1
Partition: 12 ÷ 3 = 4
2
Scale: 4 × 2 = 8
Visual Model
The Whole = 12
4
4
4
Shaded = 8
Let's Build One: 3/4 of 20
Step 1: Partition
Divide the whole (20) into equal boxes (4 boxes).
????
Step 2: Scale
How many boxes should we shade?
Real-World Scale
The Baker
A chef has 15 cups of flour. She uses 2/5 of it for bread.
Bread Flour = ?
The Architect
A building is 24 stories high. The first 3/4 of the floors are offices.
Office Floors = ?
Scaling Shapes Activity Scale Master
Fraction of a Whole: Partition and Scale
Architect Name
Protocol: 1. Draw a tape diagram. 2. Partition the whole into the number of groups (denominator). 3. Calculate how much each group is worth. 4. Scale up by the numerator.
1. Problem: \(\frac{3}{5}\) of 20
Step 1: Partition
20 ÷ 5 = _____
Put the answer in each box.
Step 2: Scale
Shade 3 boxes. Total = _____
Multiply the box value by 3.
2. Draw and Scale
\(\frac{2}{3} \times 18\)
1. Find 1/3: 18 ÷ 3 = ____
2. Find 2/3: ____ × 2 = ____
Tape Diagram
\(\frac{3}{4} \times 24\)
Write your steps below.
Tape Diagram
5.NF.B.4.A Scaling Whole Numbers Partition → Group → Scale
Product Portraits Plan Product Portraits
Lesson 7: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will multiply a fraction by a fraction (\(1/2 \times 3/4\)) by using area models to represent taking "a part of a part".
Standards
5.NF.B.4.b, ALN HLC 3
Materials
Transparencies (2 colors)
Area Model templates (square grids)
Colored markers
Key Vocabulary
Product
Area Model
Overlap
Dimensions
IM Routine
Notice and Wonder: A square partitioned vertically into 2 and horizontally into 4.
Instructional Steps
1
Warm-up: Half of a Half (5 min)
Ask: "If I have half of a pizza, and I give you half of my share, how much of the original pizza do you have?" (1/4). Draw this on the board as a square. Connect "half of half" to \(1/2 \times 1/2\).
2
Launch: The Overlap Portrait (10 min)
Model \(1/2 \times 3/4\). Draw a square. Step 1: Shade 3/4 vertically in yellow. Step 2: Shade 1/2 horizontally in blue. The area where yellow and blue overlap (green) is the product. Count the green boxes (3) and the total boxes (8). Result: 3/8.
3
Activity: Painting Products (15 min)
Students use the activity sheet to create "portraits" of multiplication problems like \(2/3 \times 1/4\). Emphasize the language: "I'm taking 2/3 of a 1/4 size piece."
4
Synthesis & Cool Down (5 min)
Ask: "What do you notice about the total number of boxes compared to the denominators?" (The total is the product of the denominators). Exit Ticket: Model \(1/3 \times 1/2\) and solve.
Visual Strength
The area model is superior to the number line for fraction multiplication because it visually explains why the product is smaller than both factors (a part of a part).
Questioning
Keep asking: "How many parts is the whole divided into now?" and "How many of those parts are shaded twice?"
Product Portraits Slides Art of Fractions
Product Portraits
Visualizing \(\text{Fraction} \times \text{Fraction}\)
A "Part" of a "Part"
When we multiply 1/2 by 1/2...
We are finding half of a half.
1/4
The "Green Zone" is the Product
The Gallery: \(2/3 \times 3/4\)
Layer 1: Shading \(3/4\)
Layer 2: Shading \(2/3\)
Overlap Boxes
6
Total Boxes
12
6/12
Artist Challenge: \(1/3 \times 1/2\)
1. Partition the square into halves.
2. Partition the square into thirds.
3. Shade the overlap. How many boxes?
DRAW ON YOUR BOARDS!
Product Portraits Activity The Portrait Gallery
Multiplying Fractions with Area Models
Artist Name
Master Protocol:
Shade one fraction vertically. Shade the other horizontally. The Green Overlap is your numerator. The Total Boxes is your denominator.
Portrait A: \(\frac{1}{2} \times \frac{3}{4}\)
1/2 High
3/4 Wide
The Count:
How many boxes overlap? ____
How many total boxes? ____
Product =
___
___
Your Exhibition: \(\frac{2}{3} \times \frac{1}{4}\)
Draw your square portrait here
Final Calculation:
2
3
×
1
4
=
___
___
5.NF.B.4.B INTERVENTION THE OVERLAP METHOD
Scaling Secrets Plan Scaling Secrets
Lesson 8: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will compare the size of a product to the size of its factors without performing calculations, based on the size of the scaling factor relative to 1.
Standards
5.NF.B.5, ALN HLC 3
Materials
Scaling Arrows (Grow/Shrink/Same)
Comparison Cards
"Greater than 1" sorting mat
Key Vocabulary
Factor
Product
Scale Factor
Resize
IM Routine
Which is greater: \(5 \times 1/2\) or \(5 \times 2\)? Why?
Instructional Steps
1
Warm-up: The Mirror (5 min)
Ask: "What happens when you multiply a number by 1?" (Stays the same). "What happens when you multiply it by 2?" (Grows). "What happens if you multiply it by 1/2?" (Shrinks). Connect these to the size of the scale factor relative to 1.
2
Launch: The Scaling Camera (10 min)
Imagine a camera lens. If you multiply by a fraction < 1, the picture shrinks. If you multiply by a fraction > 1, the picture grows. Present \(12 \times 3/4\) and \(12 \times 5/4\). "Which one will produce an answer larger than 12?"
3
Activity: The Scaling Sort (15 min)
Students use the activity sheet to sort multiplication expressions into three categories: Product < Factor, Product = Factor, and Product > Factor. They must explain their reasoning using the size of the scale factor.
4
Synthesis & Cool Down (5 min)
Ask: "Does the size of the first number matter, or just the multiplier?" (Just the multiplier). Exit Ticket: Without calculating, is \(1/2 \times 9/8\) greater or less than 1/2?
Conceptual Shift
This is a major shift for intervention students who believe "multiplication always makes numbers bigger." Use visual scaling (like on a phone) as a real-world hook.
Improper Fractions
Be sure to include fractions like 4/4 and 5/4 to help students see the "boundary" of 1.
Scaling Secrets Slides The Secret Power
Scaling Secrets
Predicting the size of the product.
The Big Question
Does multiplication
YES
2nd Grade Me
NO
Fraction Master
The Three Rules
The Shrink
× Factor < 1
Multiplying by a proper fraction makes the product SMALLER.
\(10 \times 1/2 = 5\)
The Same
× Factor = 1
Multiplying by 1 (or 4/4) keeps the product THE SAME.
\(10 \times 4/4 = 10\)
The Grow
× Factor > 1
Multiplying by an improper fraction makes it LARGER.
\(10 \times 5/4 = 12.5\)
Mental Prediction
Exp A:
\(1/2 \times 3/4\)
Exp B:
\(1/2 \times 5/4\)
The Challenge:
Don't calculate! Just point to the expression that will result in a product greater than 1/2.
A
B
Scaling Secrets Activity Scaling Inspector
Predicting Product Size: \(\times 1\), \(< 1\), \(> 1\)
Inspector Name
Multiplier < 1
Product < Factor
Multiplier = 1
Product = Factor
Multiplier > 1
Product > Factor
1. Scaling Sort
Look at the multiplier. Will the first factor shrink, grow, or stay the same?
Expression Shrink (<) Stay (=) Grow (>) \(12 \times \frac{2}{3}\)
|
|
| \(8 \times \frac{5}{5}\) |
|
|
|
| \(15 \times \frac{7}{6}\) |
|
|
|
| \(\frac{1}{2} \times \frac{1}{4}\) |
|
|
|
Mystery Mission
You are multiplying a secret number by \(\frac{9}{10}\). Will your answer be larger or smaller than your secret number?
5.NF.B.5 INTERVENTION THE SCALING SECRET: RELATION TO 1
Multiplication Masterminds Plan Multiplication Masterminds
Lesson 9: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will solve real-world word problems involving multiplication of fractions and mixed numbers using area models and equations.
Standards
5.NF.B.6, ALN HLC 3
Materials
Problem Solving Organizer
Real-world context photos
Graph paper
Key Vocabulary
Context
Area
Square Units
Mixed Number
IM Routine
Info Gap: Give one student the context (baking) and the other the numbers (\(2/3\) of \(3/4\) cup).
Instructional Steps
1
Warm-up: The Garden Patch (5 min)
Describe a garden that is 1 unit by 1 unit. If we plant carrots in 1/2 of the garden, how much area do they take? (1/2 sq unit). What if we only plant them in 1/2 of the 1/2? Connect to \(1/2 \times 1/2\). Area is the perfect context for fraction multiplication.
2
Launch: The Baking Crisis (10 min)
Problem: A recipe calls for \(3/4\) cup of sugar. You want to make only \(1/2\) of the recipe. How much sugar do you need? Model how to represent this using the "Product Portrait" method from Lesson 7, but within a "cup" or "pan" context.
3
Activity: Mastermind Missions (15 min)
Students work on complex multi-step problems on the activity sheet. Example: "A painter uses 2/3 of a gallon to paint 1/4 of a wall. How many gallons for the whole wall?" Emphasize drawing the model first to "see" the problem before writing the equation.
4
Synthesis & Cool Down (5 min)
Discuss: "When you read a word problem, what words tell you that you might need to multiply fractions?" (of, area, a part of a part). Exit Ticket: Solve one word problem from the "Challenge" section.
HLC 3 Integration
This lesson applies the conceptual models from Lessons 6-8 to "messy" real-world situations, which is a key goal of ALN's high-leverage concept for multiplication.
Language Support
For ELs or students with language processing needs, provide sentence stems: "I know I need to multiply because I am taking a part of..."
Multiplication Masterminds Slides Final Mission
Multiplication Masterminds
Solving real-world puzzles using the power of
The Chef's Secret
A recipe calls for 3/4 cup of milk. You only want to make 2/3 of the recipe.
How much milk do you need?
6/12
2/3 of 3/4 = 6/12
The Landscape Plan (Area)
A rectangular garden is 1/2 mile long and 3/5 mile wide.
Remember Area Formula:
Length × Width
3/10 sq miles
Solve It!
Problem 1
A painter has 3/4 of a gallon of paint. He uses 1/2 of it. How much paint did he use?
DRAW THE MODEL
Problem 2
A rectangular piece of fabric is 2/3 yards by 4/5 yards. What is the area?
WRITE THE EQUATION
Multiplication Masterminds Activity Mastermind Missions
Solving Fraction Word Problems
Mastermind Name
1
The Great Garden Patch
A small community garden is a rectangle. It is 3/4 of a block long and 1/2 of a block wide. What is the area of the garden in square blocks?
Equation:
Final Answer:
Draw Area Model Portrait Here
2
The Half-Batch Baker
A giant cookie recipe calls for 5/6 of a bag of chocolate chips. If you only want to make 1/3 of the recipe, how many chocolate chips do you need?
Equation:
Final Answer:
Draw Tape Diagram or Model Here
Expert Mission
Explain in your own words why multiplying \(3/4 \times 1/2\) gives you a smaller answer than both of the fractions you started with.
5.NF.B.6 INTERVENTION FRACTION FRONTIERS MISSION LOG
Dividing Units Plan Dividing Units
Lesson 10: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will divide a unit fraction by a non-zero whole number (\(1/3 \div 4\)) by using visual models to show how one part is partitioned into smaller sub-parts.
Standards
5.NF.B.7.a, ALN HLC 4
Materials
Fraction strips
Clay or playdough
Area model cards
Key Vocabulary
Unit Fraction
Partition
Sub-part
Quotient
IM Routine
Notice and Wonder: A rectangle with 1/3 shaded, then that 1/3 split into 4 pieces.
Instructional Steps
1
Warm-up: Sharing the Share (5 min)
Ask: "If I have 1/2 of a pizza and I want to share it with 2 friends, what happens to that piece?" (It gets smaller). Model this with a circle. "How much of the whole pizza does each friend get?" (1/4).
2
Launch: The "Unit" Split (10 min)
Present \(1/3 \div 4\). Draw a square. Partition into 3 vertical pieces (thirds). Shade one. "Now, 4 people want to share this 1/3. What should we do?" Partition horizontally into 4 rows. "Look at our shaded piece. It's now 4 tiny squares. Each person gets ONE of those tiny squares. How many tiny squares in the whole?" (12). So \(1/3 \div 4 = 1/12\).
3
Activity: Tiny Piece Task (15 min)
Students use the activity sheet to model division like \(1/2 \div 5\) and \(1/4 \div 2\). Focus on the "imaginary lines": "We only need to give someone a piece of the shaded part, but we have to imagine the lines going through the whole square to find the new name of the piece."
4
Synthesis & Cool Down (5 min)
Ask: "Does dividing a fraction by a whole number make the pieces bigger or smaller?" (Smaller). Exit Ticket: Solve \(1/5 \div 3\) using a quick area model.
HLC 4 Strategy
ALN HLC 4 emphasizes the relationship between multiplication and division. Help students see that \(1/3 \div 4\) is the same as \(1/4\) of \(1/3\).
Common Error
Students often just multiply the numbers and get a whole number (e.g., \(1/3 \div 4 = 12\)). The visual model is vital to show that we are starting with a small piece and making it even smaller.
Dividing Units Slides Expedition 10
Dividing Units
Splitting small pieces into even
The Shrinking Piece
Start with
1/3
Split by
4
Result
1/12
"When you divide a fraction by a whole number,
Visual Split: \(1/2 \div 3\)
1
Start with one-half of the whole square.
2
Cut that half into 3 equal pieces.
3
Each person gets 1/6 of the whole!
1/6
The Pizza Story
"There is 1/4 of a pepperoni pizza left in the fridge. 2 roommates decide to share it equally. What fraction of the whole pizza does each person get?"
1/8
Each person gets 1/8.
Dividing Units Activity The Tiny Partition
Unit Fraction \(\div\) Whole Number
Explorer Name
Expert Strategy:
1. Draw the unit fraction (vertical). 2. Split it horizontally by the divisor. 3. Count the total pieces in the whole square to find the new name!
1
Problem: \(\frac{1}{2} \div 4\)
One half split into four parts
Observation Log:
How many tiny pieces are in the shaded half? ____
If we split the other half too, how many total tiny pieces in the whole square? ____
Result =
1
___
2
Your Mission: \(\frac{1}{3} \div 3\)
Draw your Model
Final Answer:
"I know this answer is correct because when I start with 1/3 and share it with 3 people, each person gets a piece that is 3 times smaller than a third."
5.NF.B.7.A INTERVENTION FRACTION FRONTIERS MISSION LOG
Whole Dividers Plan Whole Dividers
Lesson 11: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will divide a whole number by a unit fraction (\(3 \div 1/4\)) by using visual models to show how many "parts" of a certain size fit into the wholes.
Standards
5.NF.B.7.b, ALN HLC 4
Materials
Unifix Cubes
Paper Strips (for number lines)
"How Many Fit?" Task Cards
Key Vocabulary
Quotient
Divisor
Unit Fraction
Fit
IM Routine
Number Talk: How many 1/2s are in 4? How many 1/4s are in 4?
Instructional Steps
1
Warm-up: The Whole Split (5 min)
Ask: "If I have 2 dollars and I want to change them for quarters (1/4 of a dollar), how many quarters do I get?" (8). Connect this to \(2 \div 1/4 = 8\). "Division means: How many of this size fit into that size?"
2
Launch: The Measurement View (10 min)
Present \(3 \div 1/2\). Draw 3 separate circles (or rectangles). "We want to see how many halves fit into these 3 wholes." Cut each circle in half. "Count them: 1, 2, 3, 4, 5, 6." Result: 6. "Wait, multiplication made it smaller, but division is making it BIGGER? Why?" (Because we are using a tiny unit to measure a big whole).
3
Activity: The Fitting Room (15 min)
Students use the activity sheet to model division like \(4 \div 1/3\). Emphasize: Draw the whole number of boxes first. Then, partition EACH box into the fractional unit. Count every single piece.
4
Synthesis & Cool Down (5 min)
Ask: "What is the pattern between the numbers in the problem and the answer?" (Whole number times denominator). Exit Ticket: Solve \(5 \div 1/4\) using a quick tape diagram.
HLC 4 Focus
Students must understand the "size of pieces" concept. Dividing by 1/4 is measuring with a smaller ruler, which results in a larger count.
Questioning
Always ask: "Are you splitting one thing, or are you seeing how many of one thing fit into another?"
Whole Dividers Slides Expedition 11
Whole Dividers
How many tiny parts can fit into a
The "Fitting" Question
Total Space
3
Size of Item
1/4
How many fit?
12
"When you divide by a tiny fraction,
Visual Model: \(2 \div 1/3\)
1
Draw 2 wholes (large boxes).
2
Cut EACH whole into thirds.
3
Count ALL the pieces!
1
2
3
4
5
6
The "Fit" Check
Problem:
4 ÷ 1/2
"How many halves fit into 4?"
1
2
3
4
Final Count: 8
Whole Dividers Activity The Fitting Room
Whole Number \(\div\) Unit Fraction
Tailor Name
The Golden Rule:
To see how many unit fractions "fit" into a whole number, draw the wholes, split each one into parts, and count every single piece.
1
Problem: \(3 \div \frac{1}{2}\)
Draw 3 wholes and split each into halves:
The Fitting Log:
There are ____ halves in 3.
Result = ____
2
Problem: \(2 \div \frac{1}{5}\)
Work Area: Draw your wholes and split them
Total Quotient
____
"I know this is right because each whole has 5 fifths, and I have 2 wholes. \(5 \times 2 = \text{my answer.}\)"
5.NF.B.7.B INTERVENTION THE FITTING ROOM LOG
Frontier Finale Plan Frontier Finale
Lesson 12: Fraction Frontiers Sequence
Tier 2 Intervention
Learning Objective
Students will solve complex multi-step word problems involving all four operations with fractions, choosing the correct operation based on context and visual models.
Standards
5.NF.A, 5.NF.B, ALN HLC 1-4
Materials
"Frontier Final" Scavenger Hunt
Comprehensive Fraction Toolset
Reflection Journal
Key Vocabulary
Operation
Strategy
Efficient
Verify
IM Routine
Sorting Station: Sort 4 problems into addition, subtraction, multiplication, and division buckets.
Instructional Steps
1
Warm-up: The Ultimate Choice (5 min)
Present a simple scenario (sharing pizza). Ask: "If I have 2 pizzas and 4 friends, what operation is it?" (2 ÷ 4). "If I have 1/2 of a pizza and I want 1/4 more?" (1/2 + 1/4). Briefly review the "keywords" vs "contextual meaning" of each operation.
2
Launch: Mission Control (10 min)
Introduce the Frontier Finale Scavenger Hunt. Students will work in pairs to solve 4 "Sector Missions." Each sector represents a different skill from the previous 11 lessons. They must show their work using at least one visual model for each mission.
3
Activity: The Frontier Finale (15 min)
Students complete the final activity sheet. It contains one problem for each major concept (Equivalence, Addition/Subtraction with Unlike, Fractions as Division, Fraction of a Whole, Multiplication of Fractions, Division with Unit Fractions). They must use their "Mastermind" skills to solve.
4
Synthesis & Celebration (5 min)
Review the journey. Ask: "Which model was your favorite to use?" Have students share one "Aha!" moment from the unit. Award "Fraction Frontier Certificates" for completion.
Intervention Goal
The goal is for students to feel confident enough to *choose* a model (Number Line, Area Model, Tape Diagram) that makes sense to them for any fraction situation.
Future Path
Students are now prepared for 6th grade ratios and proportional relationships, which build directly on these HLCs.
Frontier Finale Slides MISSION ACCOMPLISHED
Frontier Finale
The ultimate test of your fraction mastery.
Your Journey Log
Equivalence
Number Lines
Add & Sub
Area Models
Multiplication
Scaling
Division
Sharing & Fitting
The Master Plan: Which Operation?
"Combining two amounts together."
−
"Finding the distance between two."
×
"Taking a part of a part."
÷
"Seeing how many parts fit inside."
Mastermind Mode
Today, you won't be told which operation to use.
Fraction Freedom
You are now a
Ready for 6th Grade Ratios & Beyond!
Frontier Finale Activity Frontier Final Exam
All Operations Cumulative Review
Frontier Specialist
Your Ultimate Challenge:
Choose the operation, draw the model, and solve. Use everything you have learned on your expedition!
Sector 1: The Landscaper
A rectangular patio is 2/3 meter wide and 3/4 meter long. What is the total area of the patio?
Draw Portrait Model
Equation & Answer:
Sector 2: The Cupcake Crew
You have 1/4 of a pound of sprinkles. You want to divide them equally between 3 large cupcakes. How many sprinkles go on each?
Draw Partition Model
Equation & Answer:
Sector 3: The Relay Run
A relay track is 2 miles long. Each runner runs 1/3 of a mile. How many runners are in the race?
Draw Fitting Model
Equation & Answer:
Sector 4: The Canvas
A painter has 1/2 of a can of blue paint and 1/3 of a can of red paint. How much paint does she have total?
Draw Common Grid Model
Equation & Answer:
Expedition Reflections
Looking back at all 12 lessons, what is the most important "Scaling Secret" you learned about fractions?
5.NF.A & 5.NF.B CUMULATIVE ASSESSMENT FRACTION FRONTIERS FINAL MISSION COMPLETE
