ALEX Lesson Plan

     

How Far Can You Leap?

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  This lesson provided by:  
Author:Tim McKenzie
Organization:UAB/UABTeach
The event this resource created for:CCRS
  General Lesson Information  
Lesson Plan ID: 33080

Title:

How Far Can You Leap?

Overview/Annotation:

This lesson will allow students to become familiar with the concept of unit rate. Through an open investigation students will develop methods to find unit rate with a table, equivalent ratios, or an equation. This is a lesson to be used as part of a unit with "Painter Problems" and "How Big Should It Be?"

This is a College- and Career-Ready Standards showcase lesson plan.

 Associated Standards and Objectives 
Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 6
1. Use appropriate notations [a/b, a to b, a:b] to represent a proportional relationship between quantities and use ratio language to describe the relationship between quantities.
Unpacked Content
Evidence Of Student Attainment:
Students: Given contextual or mathematical situations involving multiplicative comparisons.
  • Communicate the relationship of two or more quantities using ratio language.
Teacher Vocabulary:
  • Ratio
  • Ratio Language
  • Part-to-Part
  • Part-to-Whole
  • Attributes
  • Quantity
  • Measures
  • Fraction
Knowledge:
Students know:
  • Characteristics of additive situations.
  • Characteristics of multiplicative situations
Skills:
Students are able to:
  • Compare and contrast additive vs. multiplicative contextual situations.
  • Identify all ratios and describe them using "For every…, there are…"
  • Identify a ratio as a part-to-part or a part-to whole comparison.
  • Represent multiplicative comparisons in ratio notation and language (e.g., using words such as "out of" or "to" before using the symbolic notation of the colon and then the fraction bar. for example, 3 out of 7, 3 to 5, 6:7 and then 4/5).
Understanding:
Students understand that:
  • In a multiplicative comparison situation one quantity changes at a constant rate with respect to a second related quantity. -Each ratio when expressed in forms: ie 10/5, 10:5 and/or 10 to 5 can be simplified to equivalent ratios, -Explain the relationships and differences between fractions and ratios.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.1.1: Define quantity, fraction, and ratio.
M.6.1.2: Identify the units or quantities being compared.
Example: Read 2/3 as 2 out of 3.
M.6.1.3: Write a ratio in appropriate notation;[a/b, a to b, a:b].
M.6.1.4: Draw a model of a given ratio or fraction.
M.6.1.5: Identify the numerator and denominator of a fraction.

Prior Knowledge Skills:
  • Compare two fractions with the same numerator or the same denominator by reasoning about their size.
  • Addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Label numerator, denominator, and fraction bar.
  • Recognize fraction 1 as the quantity formed by 1 part when a whole is partitioned into b equal parts.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.1 Demonstrate a simple ratio relationship using ratio notation given a real-world problem.


Mathematics
MA2019 (2019)
Grade: 6
2. Use unit rates to represent and describe ratio relationships.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical situations involving multiplicative comparisons,
  • Use unit rate to solve missing value problems (e.g., cost per item or distance per time unit).
  • Use rate language to explain the relationships between ratio of two quantities as non-complex fractions and the associated unit rate of one of the quantities in terms of the other.
Teacher Vocabulary:
  • Unit rate
  • Ratio
  • Rate language
  • Per
  • Quantity
  • Measures
  • Attributes
Knowledge:
Students know:
  • Characteristics of multiplicative comparison situations.
  • Rate and ratio language.
  • Techniques for determining unit rates.
  • To use reasoning to find unit rates instead of a rule or using algorithms such as cross-products.
Skills:
Students are able to:
  • Explain relationships between ratios and the related unit rates.
  • Use unit rates to name the amount of either quantity in terms of the other quantity flexibly.
  • Represent contextual relationships as ratios.
Understanding:
Students understand that:
  • A unit rate is a ratio (a:b) of two measurements in which b is one.
  • A unit rate expresses a ratio as part-to-one or one unit of another quantity.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.2.1: Define unit rate, proportion, and rate.
M.6.2.2: Create a ratio or proportion from a given word problem.
M.6.2.3: Calculate unit rate by using ratios or proportions.
M.6.2.4: Write a ratio as a fraction.

Prior Knowledge Skills:
  • Recall basic multiplication facts.
  • Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Recognize a fraction as a number on the number line.
  • Label numerator, denominator, and fraction bar.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.1 Demonstrate a simple ratio relationship using ratio notation given a real-world problem.


Mathematics
MA2019 (2019)
Grade: 6
3. Use ratio and rate reasoning to solve mathematical and real-world problems (including but not limited to percent, measurement conversion, and equivalent ratios) using a variety of models, including tables of equivalent ratios, tape diagrams, double number lines, and equations.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given contextual or mathematical situations involving ratio and rate (including those involving unit pricing, constant speed, and measurement conversions),
  • Represent the situations using a variety of strategies (tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes) in order to solve problems, find missing values on tables and interpret relationships and results.
  • Change given rates to unit rates in order to find and justify solutions to problems.
Given contextual or mathematical situations involving percents,
  • Understand the relationship between ratios, fractions, decimals and percents.
  • Interpret the percent as rate per 100.
  • Solve problems and justify solutions when finding the whole, given a part and the percent.
  • Solve problems and justify solutions when finding the part, given the whole and the percent.
  • Solve problems and justify solutions when finding percent, given the whole and the part.
Teacher Vocabulary:
  • Rate
  • Ratio
  • Rate reasoning
  • Ratio reasoning
  • Transform units
  • Quantities
  • Ratio Tables
  • Double Number Line Diagram
  • Percents
  • Coordinate Plane
  • Ordered Pairs
  • Quadrant I
  • Tape Diagrams
  • Unit Rate
  • Constant Speed
Knowledge:
Students know:
  • Strategies for representing contexts involving rates and ratios including. tables of equivalent ratios, changing to unit rate, tape diagrams, double number lines, equations, and plots on coordinate planes.
  • Strategies for finding equivalent ratios,
  • Strategies for using ratio reasoning to convert measurement units.
  • Strategies to recognize that a conversion factor is a fraction equal to 1 since the quantity described in the numerator and denominator is the same.
  • Strategies for converting between fractions, decimals and percents.
  • Strategies for finding the whole when given the part and percent in a mathematical and contextual situation.
  • Strategies for finding the part, given the whole and the percent in mathematical and contextual situation.
  • Strategies for finding the percent, given the whole and the part in mathematical and contextual situation.
Skills:
Students are able to:
  • Represent ratio and rate situations using a variety of strategies (e.g., tables of equivalent ratios, changing to unit rate, tape diagrams, double number line diagrams, equations, and plots on coordinate planes).
  • Use ratio, rates, and multiplicative reasoning to explain connections among representations and justify solutions in various contexts, including measurement, prices and geometry.
  • Understand the multiplicative relationship between ratio comparisons in a table by writing an equation.
  • Plot ratios as ordered pairs.
  • Solve and justify solutions for rate problems including unit pricing, constant speed, measurement conversions, and situations involving percents.
  • Solve problems and justify solutions when finding the whole given a part and the percent.
  • Model using an equivalent fraction and decimal to percents.
  • Use ratio reasoning, multiplication, and division to transform and interpret measurements.
Understanding:
Students understand that:
  • A unit rate is a ratio (a:b) of two measurements in which b is one.
  • A symbolic representation of relevant features of a real-world problem can provide for resolution of the problem and interpretation of the situation.
  • When computing with quantities the transformation and interpretation of the resulting unit is dependent on the particular operation performed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
M.6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation.
M.6.3.3: Calculate unit rate or rate by using ratios or proportions with or without a calculator.
M.6.3.4: Restate real-world problems or mathematical problems.
M.6.3.5: Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
M.6.3.6: Calculate missing input and/or output values in a table with or without a calculator.
M.6.3.7: Draw and label a table of equivalent ratios from given information.
M.6.3.8: Identify the parts of a table of equivalent ratios (input, output, etc.).
M.6.3.9: Compute the unit rate, unit price, and constant speed with or without a calculator.
M.6.3.10: Create a proportion or ratio from a given word problem.
M.6.3.11: Identify the two units being compared.
M.6.3.12: Define percent.
M.6.3.13: Calculate a proportion for missing information with or without a calculator.
M.6.3.14: Identify a proportion from given information.
M.6.3.15: Solve a proportion using part over whole equals percent over 100 with or without a calculator.
M.6.3.16: Form a ratio.
M.6.3.17: Convert like measurement units within a given system with or without a calculator. (Example: 120 min = 2 hrs).
M.6.3.18: Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.

Prior Knowledge Skills:
  • Recognize arithmetic patterns (including geometric patterns or patterns in the addition table or multiplication table).
    Examples: Continued Geometric Pattern by drawing the next three shapes.
  • Complete the numerical pattern for the following chart when given the rule, "Input + 5 = Output".
  • Recognize that comparisons are valid only when the two fractions refer to the same whole.
  • Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem.
  • Recognize key terms to solve word problems.
    Examples: times, every, at this rate, each, per, equal/equally, in all, total.
  • Recall basic multiplication facts.
  • Recognize equivalent forms of fractions and decimals.
  • Recognize a fraction as a number on the number line.
  • Label numerator, denominator, and fraction bar.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.6.1 Demonstrate a simple ratio relationship using ratio notation given a real-world problem.


Mathematics
MA2019 (2019)
Grade: 7
1. Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Compute a unit rate for ratios that compare quantities with different units.
  • Determine the unit rate for a given ratio, including unit rates expressed as a complex fraction.

  • Example: if a runner runs 1/2 mile every 3/4 hour, a student should be able to write the ratio as a complex fraction.)
Teacher Vocabulary:
  • Unit rate
  • Ratio
  • Unit
  • Complex fractions
Knowledge:
Students know:
  • What a unit rate is and how to calculate it given a relationship between quantities.
  • Quantities compared in ratios are not always whole numbers but can be represented by fractions or decimals.
  • A fraction can be used to represent division.
Skills:
Students are able to:
Compute unit rates associated with ratios of fractional
  • lengths.
  • Areas.
  • quantities measured in like or different units.
Understanding:
Students understand that:
  • Two measurements that create a unit rate are always different (miles per gallon, dollars per hour)
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.7.1.1: Define unit rate, proportions, area, length, and ratio.
M.7.1.2: Recall how to find unit rates using ratios.
M.7.1.3: Recall the steps used to solve division of fraction problems.

Prior Knowledge Skills:
  • Recall addition and subtraction of fractions as joining and separating parts referring to the same whole.
  • Identify two fractions as equivalent (equal) if they are the same size or the same point on a number line.
  • Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
  • Generate equivalent fractions.
  • Define quantity, fraction, and ratio.
  • Reinterpret a fraction as a ratio.
    Example: Read 2/3 as 2 out of 3.
  • Write a ratio as a fraction.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.
  • Calculate unit rate or rate by using ratios or proportions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.7.1 Calculate a unit rate (limited to whole numbers under 100).


Local/National Standards:

Math Practice Standards: 

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others. 

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning. 

Primary Learning Objective(s):

I CAN identify and develop ratios in real world situations.

I CAN identify equivalent ratios.

I CAN compare ratios in real world situations.

I CAN use equivalent ratios to find the unit rate.

Additional Learning Objective(s):

 
 Preparation Information 

Total Duration:

61 to 90 Minutes

Materials and Resources:

How Far Can You Leap? Activity Guide (found in attachments), Investigative Activity Rubric (found in attachments), Rate Exit Slip (found in attachments), Chart paper, painter's tape, Math Toolbox which includes the following: pencil, paper, graph paper, markers, scissors, glue, calculator, sticky notes

Technology Resources Needed:

Interactive whiteboard (Optional) with required software, document camera, projector, access to search engine (individually or whole group)

Background/Preparation:

The teacher must make the appropriate number of copies of the How Far Can You Leap? activity guide (found in attachments). Copies should be made so that students can work collaboratively. 

The teacher must make the appropriate number of copies of the Rate Exit Slip (found in attachments). Each student should have one.

Teacher must prepare the appropriate number of Math Toolboxes. 

Teacher must mark off a starting point for the leap. In a long hallway with square tiles works best, but this can easily be modified to do an outdoor lesson. Instead of floor tiles, the teacher can mark off feet.

The students must have knowledge of ratios. 

  Procedures/Activities: 

1. The teacher will instruct the students to search for "fastest car in the world" (if available, students can use individual devices or this can be done in small or whole group). The teacher will ask "How do we know it is the fastest car in the world?" Students will give responses, an ideal response is "because it tells us the speed." The teacher will ask the students, "How is speed displayed?" Ideal student response "Miles per hour." The teacher will introduce speed as a rate and explain that in 1 hour that car can go n miles. The teacher will introduce rate as a ratio where a unit equals 1. 

2. The teacher will ask students for other rates we use every day. Ideal student response "dollars per hour". The teacher will show the students a rate/ratio of $40 for five hours. The teacher will ask, "Is this a rate?" Ideal response, "No, because the hours is not equal to one." The teacher will pose an open-ended question "How can we get our hours down to one?" Allow students to give suggestions and strategies. Teacher will drive students to set up a table or use equivalent ratios. 

3. The teacher will introduce the activity How far can you leap? The teacher will demonstrate (or have a student demonstrate a leap), start with both feet together and jump with one foot. If needed, teacher will demonstrate the leap progression in the activity (the directions are on the activity guide). Students will begin investigation. As students are working, teacher will act as a facilitator or coach asking questions that drive understanding. 

4. Once adequate time (30-45 minutes) is given, the students will share their finding on the document camera. (If a document camera is not available students may present their work in the front of the class, this is where the students would need chart paper). As the students are sharing, the teacher is acting as the facilitator and coach asking questions that drive ratio understanding. "How do you know that ratio is equivalent to the first ratio?" "How did you know to do _______?" "Did someone do this differently?" Teacher will spawn debate on who has the best rate at jumping tiles. 

5. Toward the end of the class students will complete the Rate Exit Slip (found in attachments). 



Attachments:
**Some files will display in a new window. Others will prompt you to download.
  Assessment  

Assessment Strategies

Formal Formative Assessment: Rate Exit Slip

Formal Assessment: Using the Investigative Activity Rubric (found in attachments) teacher will evaluate students' work.

Informal Formative Assessment: As the students are working, the teacher will act as the facilitator and coach. Teacher will ask questions to evaluate students (i.e. How do you know ______?, What did you do to get that?) Teacher may pull small groups during investigation on a needs basis.

Acceleration:

The investigation has an included extension on the How Far Can You Leap? Activity Sheet (found in attachments).

Intervention:

Because this is part of a unit, teacher may develop small groups based on the Rate Exit Slip or informal questioning as part of the investigative activity.


View the Special Education resources for instructional guidance in providing modifications and adaptations for students with significant cognitive disabilities who qualify for the Alabama Alternate Assessment.