EOCCC Math Inquiry Project

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Teacher Reflections on Lesson Sequence:

Many amazing things happened during this sequence that gave way to additional thought. My filming day was very interesting as I had to quickly change my type of instruction once I recognized that students did not understand the multiplicative thinking involved in the dog question. Slowing down and having students develop a better understanding of that concept was crucial to their development throughout the sequence. Having students write similar proportional thinking questions was a great way to apply and assess their understanding of multiplicative thinking.

 

Learning Experience 4 with the Cuisenaire Rods was enjoyed by the students and myself. They really surprised me with their manipulation of the rods and how they could solve problems and assign values to the rods. This seemed to take little effort.

 

Unit rate will need to be reviewed again as part of our multiplication and division unit. This will give us additional time to reinforce the concept.

 

Our final task, the candy bar wrapper, may have been a difficult task for some of our students. I believe that some students found it challenging for many reasons: drawing the grid lines, copying the image lines onto the grid lines, or simply using their time to create a quality product. With that said, those students who accepted the challenge were immensely proud of their artwork. They couldn’t believe that they were able to make such as image.

 

Recording student work and voice was absolutely necessary with this sequence. Students in my classroom were familiar with this approach. The videos, pictures, and exit cards are an excellent tool to use for student/parent/teacher communication or for report card use. One thing that I would like to develop is a quick checklist where I can quickly assess the mathematical process or strategy that individual students may be using. In addition, an area where I can note concerns or accomplishments would also be a great asset.

Planning the Lesson Sequence

Michelle Jessup talks about what it was like to plan the lesson sequence, online, with another teacher she had only met once before.

300 Minute Block

Jessica Mulhall discusses what it was like planning a 300 minute lesson sequence.

Big Ideas

Michelle Jessup describes the "Big Ideas" of the

Lesson Sequence she and Jessica Mulhall designed

Choosing Partners

Michelle Jessup describes how she chosses who

students will be partnered with for lessons.

Focus on Proportional Reasoning

Jessica Mulhall explains why they chose to focus on proportional reasoning and how where they found resources to help support their planning.

Differentiation

Jessica Mulhall discusses ways that she differentiates in her classroom in order to meets the needs of all of her students.

Assistive Technology

Michelle Jessup describes how assistive technology is being used in her classroom for a student with a hearing impairment.

Students' Struggles

Jessica Mulhall discusses some of the ways she has seen students truggle with Proportional Reasoning.

Assessment

Michelle Jessup discusses how she and her

students created assessment criteria together.

LEARNING EXPERIENCE 1 (50 minutes)

annotated lesson sequence: Grade 6/7

Learning Goal

  • We can recognize proportional thinking in problem solving. Use a problem solving task to activate prior knowledge.

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Students share some of their thinking from the "Minds On" task

 

 

Students share some of their thinking from the "Action" task

 

Co-constructed criteria for problem solving

1. Read and reread the problem

2. Understand the problem (ex. highlight/underline the important information #'s that are important that help us know what operation to use)

3. Make a plan. What math tools can you use. Write down what you know. Can you make any connections?

4. What do you estimate the answer to be?

 

5. Put your thinking down on paper. All work on calculator needs to be recorded. Work steps should be in a logical order. Did you use a mixture of numbers, pictures and/or words?

6. Double check all calculations.

7. Is your final answer reasonable?

 

[ENLARGE]

Minds On

 

Look at this image. Think about what you see. Turn and talk to your elbow partner about what you see.

 

[ENLARGE]

Source: Paying Attention to Proportional Reasoning, p. 5

 

[Partner and Whole Group, 10 minutes]

 

 

Action

 

Which shape is more purple?

 

[ENLARGE]

Source: Paying Attention to Proportional Reasoning, p. 3

 

[Think, Pair, Share, 5 minutes]

 

Samples of Student Work

 

[ENLARGE]

 

 

Review co-constructed criteria for problem solving and working collaboratively.

 

Ask students to meet with a partner and verify their conjectures. Have a variety of manipulatives available (E.g. rulers, fraction strips, calculators, etc.).

 

[Think, Pair, Share, 5 minutes]

 

Discuss students’ conjectures by setting up a value line (Priniciples of Effective Literacy, pg. 154).

 

Pairs of students need to select which area has more purple.  Students will need to defend their position when asked,, “Which shape is more purple?”

  • As students to share their strategies, note name, and record their ideas.
  • Ask students to reflect on how their strategy was the same and/or different.
  • Ask if any student wishes to move to the other end of the value line, signifying a change in their thinking.

 

[Think, Pair, Share and Discussion, 10 -15 minutes]

 

Consolidation:

 

Ask:

  • What did proportional reasoning look like today?
  • What strategies did you use today to help you solve these problems?
  • Record information on anchor chart.
  • After class discussion, have students record their thinking in their math journal. This can be done using a book or digital app, like Explain Everything or Show Me.

 

[Individual, 15 minutes]

 

 

 

 

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LEARNING EXPERIENCE 2 (55 minutes)

annotated lesson sequence: Grade 6/7

Learning Goal

  • We can apply multiplicative thinking to solve problems. Guide students through an investigation to explore the idea of absolute versus relative thinking.

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Minds On

 

If one dog grows from 5 kg to 8 kg and another dog grows from 3 kg to 6 kg, which dog grew more?

 

 

[ENLARGE]

Source: Paying Attention to Proportional Reasoning, p.5-6

 

 

Using Think-Pair-Share, students will compare the growth of the two different dogs. Ask students to explain their thinking. Record initial responses on anchor chart of students’ additive and multiplicative thinking.

 

[Think-Pair-Share, 10 minutes]

 

Note: Both teachers involved in the project found that their students needed direct teaching in order to solve this problem. Very few students were able to solve the problem using multiplicative thinking.

 

Action

 

Review learning goal and co-constructed criteria for problem solving and working collaboratively.

 

Have students work with a partner to solve the lemonade problem.

 

[ENLARGE]

Source: Teaching Student Centered Mathematics 6 - 8, Van de Walle, pg. 214.

 

[Partners, 20 minutes]

 

Some examples of student work are shared here. You can download two class sets of student work from two different schools.

 

 

[ENLARGE]                                                    [ENLARGE]

 

 

[ENLARGE]                                                    [ENLARGE]

 

 

Consolidation

 

Conduct a Gallery Walk.  Ask students to reflect on which pairs used similar and different strategies.

Ask:

What solutions showed additive thinking? Which solutions showed multiplicative thinking?

What is a question you may have about another group’s work?

What strategy did you find most efficient?

 

Review learning goal and add new ideas or vocabulary words to anchor chart.

 

[Whole group, 15 minutes]

 

Exit Ticket

 

Parallel Task

 

 

[ENLARGE]

 

Examples of Student Responses

 

[ENLARGE]

 

[ENLARGE]

 

[ENLARGE]

 

[ENLARGE]

 

 

[Individual, 10 minutes]

 

NOTE: Students in both classes had difficulty with understanding the concpet of proportional reasoning. The teachers therefore decided to do a "Take 2". The following video and lesson plan is one teachers second lesson plan for this learning experience.

 

Learning Experience 2 - Take 2

 

Day 2 - Take 2

 

Student Pairing:

Students worked with the same partners

 

Challenges:

Have students think multiplicatively rather than additively.

Make sure questions are worded well. Be certain students understand the question.

 

Successes:

Students solved similar style questions to the dog question from the Minds On. Then they worked at creating their own question that was similar to the dog question that would allow for multiplicative/ proportional reasoning. These questions were solved and were part of our Consolidation for proportional reasoning.

 

Example of Student created Question:

 

 

Next Steps:

Get back to the sequence. Students will use their questions during the Minds On.

 

Reflection:

As teachers, we need to listen and be attentive to our students’ needs. I could have continued with the sequence, hoping that the students would “get” the concept as we carried on, but this is completely unfair to them as learners. It is acceptable for students to struggle with a concept and for teachers to realize that their approach didn’t work and to try again!

 

 

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Describing the Learning Experience

 

Setting Up Learning Experience 2

 

Minds On Task: Introduction and Sharing Solutions

 

Minds On Task: Consolidation part 1

 

Minds On Task: Consolidation part 2

 

Minds on Task: Introduction and Consolidation

 

Additive vs. Mulitplicative Thinking

Several weeks later the class revisited additive vs. multiplicative thinking and the students recalled the concepts and explained it to the teacher.

 

Action: Lemonade Problem Introduction

 

Lemonade Problem: Student Thinking

o

 

Gallery Walk - Intro

 

Gallery Walk - Student Thinking

 

Consolidation

 

Multiplicative Thinking - Take 2

 

LEARNING EXPERIENCE 3 (45 minutes)

annotated lesson sequence: Grade 6/7

Learning Goal:

  • We can use unit rate to solve proportional reasoning problems. Guide students through an investigation involving unit rate and proportional reasoning.

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Buying Apples

Minds On

 

Kellie says that if 3 apples cost her $1.50, then she should be able to buy 9 apples for $5.00. Is she correct?

 

Discuss and display various strategies that pairs of students used to find a solution. Record specific strategies.

 

Teacher should highlight the idea of unit rate.

 

[Partner and Whole Group, 10 minutes]

 

Action

 

Review learning goal.

 

Have students work with a partner to solve the following problem.

The rates for Internet use offered by three companies are shown below.

  • Company A: $6.00 for every 90 minutes of use
  • Company B: $2.75 for every 45 minutes of use
  • Company C: $3.00 for every 60 minutes of use

Which company offers the lowest rate per minute?

Source: Paying Attention to Proportional Reasoning, p.13

 

[Partners, 10 minutes]

 

Examples of Student Responses

 

 

[DOWNLOAD]

 

Consolidation

Facilitate a discussion using selected work. Have pairs of students show their thinking and explain their strategy.

Ask:

  • Convince us that your strategy led to a reasonable solution.
  • Are there any strategies related?
  • Which strategy is efficient and effective for you?
  • What have you discovered about proportional thinking while solving this problem?” Review learning goal and add new ideas.

[Whole Group, 15 minutes]

 

Exit Ticket

Choose a price for four cinnamon buns. Then choose a different number of cinnamon buns and tell how much that new number of buns would cost. Justify your answer is correct.

 

Source: Good Questions Great Ways to Differentiate Mathematics Instruction, Marian Small, p. 46.

 

*Students can use a paper handout (download), Explain Everything or another digital tool to record their thinking. Students should annotate their solution, by referring to the problem solving success criteria.

[Individual, 10 minutes]

 

[DOWNLOAD]

 

 

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LEARNING EXPERIENCE 4 (50 minutes)

annotated lesson sequence: Grade 6/7

Learning Goal:

  • We can use fractional benchmarks to think proportionally. Direct students through an investigation to establish proportional thinking with open tasks.

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Resources

Teachers found the following support documents important resources when planning these learning experiences

 

Paying Attention to Proprotional Reasoning

(download)

 

Asking Effective Questions

(download)

 

Cuisenaire Rods

 

Minds On

 

Have students work with a partner to solve the following. (Students should estimate how many rods they would need before beginning the task)

 

If the table is 6 orange Cuisenaire rods long, how long would it be if measured using a different coloured rod?

 

Adapted from Paying Attention to Proportional Reasoning, p.9

 

Have pairs of students record their findings on paper,  journal, whiteboard, Explain Everything, etc. Ask them to think about the relationship between the rods they have chosen?  Click here for a hand out of the question above.

 

Examples of Student Responses

 

 

 

 

[DOWNLOAD]

 

[Partners, 10 minutes]

 

Action

 

Review learning goal.

Give pairs of students Cuisenaire rods. Allow them to manipulate them and come to some understanding of them as individual rods and their relationship to other rods.

After a few minutes of “play,” challenge students with the following questions:

  • If the yellow rod is worth 5, what are the other rods worth?
  • Now the light green rod is worth 15, what are the other rods worth?
  • If the orange rod is worth 1, what is the yellow rod worth? (1/2)
  • Choose a value for the purple rod. What are the other rods worth?

 

Debrief

 

After each question and have students share their representations using a document camera or mirroring.

 

[Individual, Whole Group, 15 minutes]

 

Consolidation

 

Have each group of students partner with another group reflecting upon similar and different strategies and or/rods used. Ask:

  • Which one-colour rod combinations matched the length of the orange rod?
  • How did you determine the fractional part that each of these colour rods represented in relation to the whole?
  • Which rods were the easiest to figure out? The hardest to figure out? Explain.
  • What patterns or relationships did you discover between the rods?

Adapted from The Super Source, Cuisenaire Rods (Gr. 7-8); Yack in the Box

 

Review learning goal and chart new ideas.

 

[Small Group & Whole Group, 15 minutes]

 

Exit Ticket (download printable sheets)

1. “Where do you see problems similar to this? At home? At school? Outside?”

Source: Capacity Building Series, Asking Effective Questions, p. 4.

 

2. “What challenges are you still facing with proportional reasoning?”

[Individual, 10 minutes]

 

Teacher Reflection:

Students were comfortable using the Cuisenaire Rods as they had previously used them this year. Students were building on their understanding of proportional reasoning. The last question in the Action, “Choose a value for the purple rod. What are the other rods worth?” was a fantastic question for students to define and extend their understanding of proportions. Some pairs of students tried to challenge themselves by dealing with smaller numbers rather than larger numbers. They really enjoyed the hands - on tasks in this lesson.

 

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LEARNING EXPERIENCE 5 (55 minutes)

annotated lesson sequence: Grade 6/7

Learning Goal:

  • We can apply proportional thinking to produce a piece of artwork. Students will apply their learning about proportional thinking and how it relates to various mathematical strands, concepts, or big ideas.

Overall Expectation:

Grade 6

  • demonstrate an understanding of relationships involving percent, ratio, and unit rate

Grade 7

  • demonstrate an understanding of proportional relationships using percent, ratio, and rate

Specific Expectations:

Grade 6

  • represent ratios found in real-life contexts, using concrete materials, drawings,and standard fractional notation
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions, decimal numbers, and percents
  • represent relationships using unit rates

 

Grade 7

  • determine, through investigation, the relationships among fractions, decimals, percents, and ratios
  • demonstrate an understanding of rate as a comparison, or ratio, of two measurements with different units
  • solve problems involving the calculation of unit rates

Student Math Journal Entry

[ENLARGE]

 

Minds On

 

Review the learning goal.

Ask students to revisit their math journals  and anchor charts and recall new concepts learned this week about proportional thinking.

 

[Think, Pair, Share, 10 minutes]

 

Action

 

Students will work individually to enlarge or decrease their candy bar wrapper on a self-created grid  to show their proportional thinking. Using a standard 1 cm x 1 cm grid acetate to cover their candy bar wrapper will give students a starting point.

Students will need to select appropriate sized paper to create a new proportional drawing of the wrapper.

 

As they begin this activity, they should be thinking about why this is a proportional thinking problem.

 

[DOWNLOAD]

 

[Individual, 35 minutes. More time will be needed during art blocks to finish their product]

 

Consolidation

 

Ask students

“Why is the Candy Bar Problem a proportional thinking problem?”

“How did you use proportional thinking in your drawing?”

“What is the thing you liked best about this week in Math class?”

“What was the hardest part of this week in Math class?”

Adapted from Capacity Series, p. 5

 

Students should reflect on these questions using Show Me or Explain Everything App, video or iMovie, or writing in Math journal. They need to use specific details in their reflections.

 

[Individual, 15 minutes]

 

Teacher Reflections

 

Challenges:

Students have to measure precisely. They need to decide on a scale to use that will enlarge or decrease the candy bar wrapper that they chose.

Drawing the lines on a piece of paper was challenging for a few of my students so I needed to help them with that. Those students who have weaker fine motor skills did have difficulty with this task even when the grid lines were drawn for them.

It took 2 additional periods to accomplish the grid lines and the drawing. Many students worked on the colouring in spare time or at home.

 

Successes:

Those students who were determined to finish the task were very proud of their finished product. Many compliments were given to the students. They did recognize that this was another aspect of proportional reasoning.

 

 

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