Grade 5 Math Unit Planning 2017 to 2018PS 105

Unit# / Book(s) / Topic Unit 2 / Books 3 & 7 / Fraction Operations Approximate Days or Dates 40 days

Stage 1 - Identify Desired ResultsLearning OutcomesWhat relevant goals will this unit address?(must come from curriculum; include specific Common Core standards)

Use equivalent fractions as a strategy to add and subtract fractions

5.NF.1: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators.

5.NF.2: Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g, by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.

Apply and extend previous understandings of multiplication and division

5.NF.3: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.5.NF.4.A: Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)5.NF.4.B: Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.

5.NF.5: Interpret multiplication as scaling (resizing), by:5.NF.5.A: Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.5.NF.5.B: Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.

5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1

5.NF.7.A: Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.5.NF.7.B: Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.5.NF.7.C: Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?

Other

5.OA.2: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.

5.MD.2: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions to solve problems involving information presented in line plots.

Enduring UnderstandingsWhat understandings about the big ideas are desired?What misunderstandings are predictable?

Essential QuestionsWhat are the Essential Question(s) of the unit?Are there any potential cross-curricular connections during this unit?

Students will understand that...

● Fractions are numbers that have a place on a number line.

● Fractions can only be added or subtracted when the parts are alike. Just like when we add and subtract whole numbers by adding tens with tens and ones with ones, we need to add like fractional parts.

● Fraction multiplication can be understood similarly to whole number multiplication as equal groups. Thus 1/3 x 9 can be thought of as 9 groups of 1/3 or 1/3 of a group of 9.

● Dividing a whole number by a fraction (e.g., 3 / ½) is best understood as how many ½’s make up 3 wholes?

● Dividing a fraction by a whole number (e.g., ½ / 3 is best understood as ½ shared in 3 equal groups.

Related misconceptions…

● Students have a hard time with the idea that dividing a number by a fraction makes the number bigger, whereas multiplying by a fraction (less than 1) makes the number smaller.

Essential Question:

● How is computing with fractions similar to and different from computing with whole numbers?

Cross-curricular connections…

Knowledge:What knowledge will students acquire as a result of this unit?

Skills:What skills will students acquire as a result of this unit?

Students will know...

● There are an infinite number of ways to express any fraction, but it is customary to use the simplest form when writing final answers.

● Similarly, we generally change improper fractions to mixed numbers when writing a final answer.

● Multiplying a number by a fraction less than 1 decreases the number.● Multiplication of fractions is commutative. Therefore, taking 2/3 of 9 is

equivalent to taking 9 groups of 2/3.

Students will be able to…

● Compare and order fractions and mixed numbers.● Add fractions and mixed numbers with different denominators.● Subtract fractions and mixed numbers with different denominators.● Estimate sums and differences of fractions and mixed numbers.● Solve complex story problems involving adding and subtracting fractions

or mixed numbers.● Create area (array) models for multiplying fractions, especially for

fractions of fractions.● Find a fraction of a whole number or a whole number times a fraction.● Multiply fractions and mixed numbers using models, the distributive

property or the basic algorithm for multiplying proper fractions.● Divide fractions by whole numbers without algorithms.● Divide whole numbers by fractions without algorithms.

Stage 2 – Assessment EvidenceEvidenceThrough what evidence (work samples, observations, quizzes, tests, journals or other means) will students demonstrate achievement of the desired results? Formative and summative assessments used throughout the unit to arrive at the outcomes.

Student Self-AssessmentHow will students reflect upon or self-assess their learning?

Pre-Assessment: Beginning of Year Assessment questions 4, 5, 17, 18, 19, 21, 28, 30 and Pearson Unit 3 #6 and 15 and Pearson Unit 7 #12 and 19.

Book 3:

Benchmark 1: Add fractions with unlike denominators.Quiz 2

Benchmark 2: Subtract fractions with unlike denominators.Quiz 2

Benchmark 3: Represent data including fractions on a line plot and solve addition and subtraction problems about the data.Resource Master A26

Book 7:

Benchmark 1: Multiply fractions, mixed numbers, and whole numbers.Quiz 1

Benchmark 2: Compare the size of the factors and the size of the product and explain their relationship. Quiz 1

Benchmark 3: Divide a unit fraction by a whole number and a whole number by a unit fraction. Quiz 1

Benchmark 4: Solve division problems with two whole numbers resulting in a fraction or a mixed number.

Pearson Unit 3 Assessment Questions: 1-3, 5-8, 11, 13-16, 18-21Pearson Unit 7 Assessment Questions: 2, 3, 4, 6, 8, 10, 12, 13, 22Plus additional questions TBD

Stage 3 – Learning Plan# Content Goal Lesson Notes/Planned Differentiation Additional Resources or Math Centers

Unit Note: We are going to use the CGI approach to understanding fractions to supplement this unit. The suggested CGI lessons are numbered from CGI-1 to CGI-7.

Extending Children's Mathematics: Fractions & Decimals: Innovations In Cognitively Guided Instruction by Emson and Levi

Fluency Ideas EngageNY Sprints that are recommended for this unit:● Module 3: 2, 3, 5, 7, 9, 10, 12, 14, 15● Module 4: 6, 14, 18, 30● Module 5: 3, 7, 18● Module 6: 19, 20, 23

Test Prep Ideas State Test released items on fractions:● 2015: 51, 52, 53, 55, 59● 2016: 47, 49, 52, 54● 2017: 46, 47, 49, 52, 54, 55

Practice Problem Sets: 4, 5, 7, 8, 14, 16, 17, 22, and 236

sessionsBook 3:Investigation 1:Comparing and Ordering Fractions

This first investigation reviews several important grade four concepts--comparing and ordering fractions and finding equivalent fractions.

1.1 Reviewing Prior Knowledge of Fractions

Do this lesson as written, but be sure to review tape diagrams as a model for finding a fraction of a whole number.

1.2 Reviewing Equivalent Fractions

This lesson is written to extend into the next day’s lesson, but many students will be able to complete all 5 SAB pages today. Introduce both types of pages during the mini-lesson.

1.3 Ordering Fractions Reverse the intended order of this lesson. Start with the discussion, “How Do You Know it’s ½?”, then ask students to work with a partner to order the Fraction Cards, and end with a sharing of student strategies used to order fraction. The emphasis should be on ways to use number sense to compare and order fractions, NOT merely on getting common denominators and certainly NOT by cross multiplying.

1.4 Comparing Fractions Do this lesson as written, but be sure to have something extra available for early finishers.

1.5/1.6 Comparing and Ordering Fractions

“In Between” is an excellent game. Try having half the class play it while the other half works on SAB pp. 142, 143, 147, and 148 and then switch. This is set up to continue for two days. Your class may only need one day, so see how it goes. If part of your class needs two days, provide the other part with some some more advanced problems on the second day, including the Extension activity on TG p. 63.

● In Between● SAB pages

8 sessions

Book 3:Investigation 2:

Our recommendation is to replace sessions 2.1, 2.2, and 2.3 (adding fractions with clocks) with the four CGI lessons detailed below.

Adding and Subtracting Fractions

However, if you would like to try an alternative approach this year, go ahead and use the fractions with clocks set of lessons instead and let us know how it works.

CGI-1 Equal Sharing Problems and equivalent fractions

Spend one day (CGI-1) reviewing equal sharing problems from the CGI book. Focus on problems that may lead students to have to add fractions with different denominators. Try these problems:

__children want to share __ sub sandwiches so that each one gets the same amount. How much should each person get?

(2, 1 ¾) (3, 2 ¼) (2, 1 7/8) (3, 3 ¾) (3, 3 ¼)

CGI book, chapters 1 and 2 (see pages 32-35 for advice on problem selection)

CGI-2 Adding and Subtracting fractions and mixed numbers with friendly denominators

Try these problems today, asking students to use visual models to figure them out, rather than algorithms they may have learned in outside math instruction.

Sara likes to eat __ sandwich for snack and __ sandwich for lunch. If Sara is going to eat her snack and lunch at school, how many sandwiches does she need to bring to school?

(1/2, 3/4) (3/4, 1 ½) (1/2, 3/8) (1/2, 7/8) (3/8, ¾) (3/4, 1 1/8)

Yvette had __ jars of jam. After she made sandwiches for her friends, she had __ jars of jam left. How much jam did she use?

(3, 2 ½) (3 ¼, 2 ½) (4, 2 ½) (4, 2 ¾) (3, 2 7/8)

CGI book, chapter 8 (see pages 209-211 and 218-221 for advice on problem selection)

CGI-3 Adding and Subtracting fractions and mixed numbers with friendly denominators

Today and tomorrow move from story problems to equations. Use all of the Addition and Subtraction Equations on pages 210 to 211. See if students can do these without manipulatives or drawings, but allow them to use these models if necessary. The goal is to move students to using equivalent fractions to create common denominators.

Ditto

CGI-4 Adding and Subtracting fractions and mixed numbers with friendly denominators

Continue the work from yesterday. Today students should focus on the last set of problems, where both denominators need to be changed. If these prove difficult, let students go back to drawings or fraction strips and encourage them to discover the algorithm, rather than telling them today. They should begin to discover that when one denominator is not a multiple of the other denominator, they can use the product of the denominators as their common one. Don’t worry about LCD. It is not required for fifth grade.

Ditto

2.4 Fractions on Number Lines

Do this lesson as suggested. We are not formally using Quiz 1, but you can use it for informal assessment.

2.5/2.6(2 days)

Fraction Track Game Do this two-day lesson as suggested. The idea is to play this game WITHOUT the use of the standard algorithm for addition of fractions, but instead to use what they have learned about Equivalent Fractions.

2.7 Subtracting Fractions

Do this lesson as suggested, including giving the assessment at the end.

6 sessions

Book 3:Investigation 3:Adding and

As you come to the end of the work on addition and subtraction of fractions, there are some great story problems to use as a supplement.See EngageNY Module 3 Problem Sets 7, 8, and 15. (Note that

http://www.k-5mathteachingresources.com/5th-grade-number-activities.html

Subtracting Mixed Numbers

RDW stands for Read, Draw, Write, which is a great way to help students approach story problems. See Module 1 pages ix-x for a formal definition.)

3.1 Fraction Track to 2 Do this lesson as written with students splitting their time between the game and the addition and subtraction practice on the SAB pages.

● Fraction Track to 2● SAB pages

3.2 Adding and Subtracting Fractions

This lesson is a continuation of 3.1. By this time, work to ensure that all the students understand formally how to add and subtract fractions by using equivalent fractions to get common denominators.

● Fraction Track to 2● SAB pages

3.3 Addition Compare with Fractions

Do this lesson as written with students splitting their time between the game and the addition and subtraction practice on the SAB pages.

● Addition Compare with Fractions● SAB pages

3.4 Fractions on a Line Plot

Today everyone should complete the Grasshopper Lengths activity and then divide the remaining time between the game and the additional SAB pages. Be sure to save enough time for the discussion of addition and subtraction of mixed numbers at the end.

● Addition Compare with Fractions● SAB pages

3.5/3.6(2 days)

Adding and Subtracting Mixed Numbers

These final two days of the unit consist of a number of SAB pages, a quiz, and an assessment. Use the time to support any students still needing help with addition and subtraction. Be sure to have extra challenge activities available for early finishers, including playing Addition Compare with 3 Fractions as described on TG p. 168.

QUIZ 2 plus assessment page A26 (M-U-A)

13sessions

Book 7: Investigation 1:Multiplying and Dividing Fractions

CGI-5 Multiple Groups Problems

Spend three days working on multiple groups problems from the CGI book. Use equations to summarize student results so that they understand they are multiplying fractions by whole numbers. And highlight students’ strategies that use multiplicative relationships to solve these problems (For example, listen for students who say things like, “I know 4 groups of ¾ is 3 so 20 groups of ¾ would be 5 times 3 or 15.”

Start today with two problems: It takes ¾ pound of clay to make a bowl. How much clay is needed to make 12 bowls? It takes 3/8 cup of sugar to make a loaf of bread. How much sugar would you need for 16 loaves?

CGI book, chapter 3 (see pages 69-71 for advice on problem selection)

For homework and/or last ten minutes of classwork, give a sheet reviewing the skills of addition and subtraction of fractions.

CGI-6 Multiple Groups Problems

Start off today with a number talk using the open number sentences on page 67 of the CGI book. Then do this problem:

Eve’s Gecko eats 2/7 jar of baby food a day. How many jars should she buy to last for two weeks? If she buys a case of 24 jars, how many days would that last?

CGI-7 Multiple Groups Problems

Today skip the contexts and solve a variety of open number sentences such as these: ¾ x 14 = ? ¾ x ? = 12 ¾

1.1 Multiplying a Fraction by a Whole Number

This lesson is fine as written. It continues the work begun with the CGI Multiple Groups problems.

1.2 Multiplying a Whole Number by a Fraction

NOTE: The Multiple Groups problems of the past few days involved finding a whole number of groups of a fraction. Today it is the opposite, finding a fraction of a group. For these types of problems, the bar model should be used.For 4/5 of 20, it would look like this:

4 4 4 4 41.3 Multiplying Whole

Numbers by Fractions and Mixed Numbers

This lesson is fine as written. Standard algorithms known by students should only be done to check work. Models should still be drawn.

1.4 Multiply fractions or Mixed Numbers

This lesson is fine as written.

1.5 Multiplying Fractions by Fractions

This lesson is fine as written, and is likely easier for students to visualize than an area model. Note that the area model is introduced in lesson 1.7.

1.6 A Rule for Multiplying Fractions

It is OK to use the algorithm for fraction times a fraction, but continue to have students practice creating models, because they are difficult and absolutely necessary for the state test. Students who already knew the rule should be focusing on understanding and explaining WHY the rule works.

1.7(2 days)

Using Arrays for Multiplying Fractions

This is a tricky model that appears often on the state test, so take an extra day to practice this.

1.8 Multiplying Fractions and Multiplying Mixed Numbers

This lesson is fine as written, but be sure that students include array models with their work.

1.9 Dividing a Whole Number by a Fraction

These types of problems are essentially CGI Multiple Groups problems with the number of groups unknown. That is how they should be solved. Do NOT teach or allow the use of the “Invert and Multiply” algorithm. It is completely unnecessary for grade 5 and harmful to students’ thinking and understanding of the meaning of division of fractions.

1.10 Dividing a Fraction by a Whole Number

Again, emphasize the use of models to solve these problems and promote understanding.

Give QUIZ 1 (Book 7) today

1.11 Dividing with Fractions

This lesson is fine as written.

Extra Day Fraction Computation Review

Give students a day to practice solving fraction story problems with all four operations mixed up. Note that students often have trouble identifying the correct operation to use when solving fractions problems.

Test Unit Assessment Pearson Unit 3 Assessment Questions: 1-3, 5-8, 11, 13-16, 18-21Pearson Unit 7 Assessment Questions: 2, 3, 4, 6, 8, 10, 12, 13, 22Plus additional questions TBD

Post-Unit ReflectionConsiderations CommentsRequired Areas of Study: Was there alignment between outcomes, performance assessment and learning experiences?

There wasn’t enough preparation for the story problems students had to solve in the unit test. We took out two or three of the harder questions from the unit test and next year we want to provide more challenging mixed review of story problems for students to solve. There are 3 days built in for addition and subtraction, but we need a few days at the end for story problems with all four operations. Look in Engage for this…maybe even the mid-module review.

Adaptive Dimension:Did I make purposeful adjustments to the curriculum content (not outcomes), instructional practices, and/or the learning environment to meet the learning needs and diversities of all my students?

For struggling students:

For students who need a challenge:

Suggested Changes:How would I do the unit differently next time?