BACKWARDS DESIGN BASICS • Begin with the end in mind • Develop a clear understanding of where you want to go • Map out the steps to get there www.arps.org STAGES TO BACKWARD DESIGN Stage 1: Identify Desired Results • What do you want the students to know and be able to do? • Standards Stage 2: Determine Acceptable Evidence of Learning • Assessment tools that will act as evidence for student understanding • Formative to drive instruction Stage 3: Design Learning Experiences and Instruction • Unit/lesson plans that include activities, plans for meeting needs for all students and instructional tools BUILDING ASSESSMENT QUESTIONS • Types of Assessments • Power of the Question • Common Core Aligned Tasks Oakland Schools Website Summative Assessment Formative Assessment Measure full range of CCSS Repository of tools available to teachers Computer Adaptive Testing for precision to support quick adjustment and Timely results differentiated instruction Engage Institutions of Higher Education Help define student performance along to ensure achievement standards reflect the CCSS learning progressions college and career readiness Concrete strategies for immediate Scale scores help inform growth mode feedback loops Interim Benchmark Assessment Allow for finer grain of measurement (e.g., end of unit) Inform teachers if students on track to be proficient on summative assessments Multiple opportunities for students to participate Scale scores help inform growth model Oakland Schools Website Are you assessing content or practice? OR BOTH? Maine West Department of Education webiste POWER OF THE QUESTION Understand and use ratios, proportions and percents in a variety of situations. –NJ Core Curriculum Content Standards for Mathematics Grade 7, 4.1.A.3 1. 50% of 20: 2. 67% of 81: 3. Shawn got 7 correct answers out of 10 possible answers on his science test. What percent of questions did he get correct? 4. J.J. Redick was on pace to set an NCAA record in career free throw percentage. Leading into the NCAA tournament in 2004, he made 97 of 104 free throw attempts. What percentage of free throws did he make? 5. J.J. Redick was on pace to set an NCAA record in career free throw percentage. Leading into the NCAA tournament in 2004, he made 97 of 104 free throw attempts. In the first tournament game, Redick missed his first five free throws. How far did his percentage drop from before the tournament game to right after missing those free throws? 6. J.J. Redick and Chris Paul were competing for the best free-throw shooting percentage. Redick made 94% of his first 103 shots, while Paul made 47 out of 51 shots. Which one had a better shooting percentage? In the next game, Redick made only 2 of 10 shots while Paul made 7 of 10 shots. What are their new overall shooting percentages? Who is the better shooter? Jason argued that if Paul and J.J. each made the next ten shots, their shooting percentages would go up the same amount. Is this true? Why or why not? Hand Out Driven By Data What are the differences between these six assessments of the same standard? What conclusions do you draw about assessments when looking at these examples? POWER OF THE QUESTION Driven By Data 1. What is the main idea? 2. This story is mostly about: A. Two boys fighting B. A girl playing in the woods C. Little Red Riding Hood’s adventures with a wolf D. A wolf in the forest 3. This story is mostly about: A. Little Red Riding Hood’s journey through the woods B. The pain of losing your grandmother C. Everything is not always what it seems D. Fear of wolves INK WHILE YOU THINK: WHAT ARE THE DIFFERENCES BETWEEN THESE THREE QUESTIONS? WHICH QUESTION IS MORE RIGOROUS? Driven By Data POWER OF THE QUESTION ACTIVITY Why? Analyze the questions based on rigor How? Silently look through the sample test Pick out 2-3 questions that could be asked in a more rigorous way Pick 2- 3 questions that are rigorous for third grade. Why? Be ready to discuss your questions Concepts to Remember: • Options define the rigor of multiple choice questions • Rubric defines the rigor in open ended questions • Good assessments will combine multiple choice forms to achieve the best measure of mastery POWER OF THE QUESTION Driven By Data Page 21 ASSESSING CONTENT AND PRACTICE • Separating the practices from the content is not helpful and is not what the standards require. • The practices do not exist in isolation; the vehicle for engaging in the practices is mathematical content. • The K-6 math standards already have implied practice standards embedded ORIGINAL TO BETTER 3.OA.3 QUALITIES OF ALIGNED TASKS • Intentionally selected to elicit one or more particular practices for specified content • Each targeted practice is observable in the student’s response, directly in student work or indirectly through an incorrect answer to a problem in which the practice would have made a correct answer much more likely. • It must be unlikely or impossible to earn full credit on the task without engaging in the practice. PERSEVERANCE TASKS MP.1 Make sense of problems and persevere in solving them • Word problems involving ideas that are currently at the forefront of the student’s developing mathematical knowledge (e.g., multiplication in Grade 3 or addition or subtraction of fractions in Grade 4). • Are designed to take a typical student a long time to solve; • Require a large number of routine and fairly easy steps; • May involve steps which in turn lead to a more difficult problem • May have a rubric that awards increasing points to each part There will be 58 people at a breakfast and each person will eat 2 eggs. How many cartons of eggs will be needed for the breakfast. There are 12 eggs in each carton. ABSTRACT & QUANTITATIVE REASONING TASKS MP.2 Reason abstractly and quantitatively Contextual problems in which the student can gain insight into the problem, and earn points on the task, by relating the algebraic form of an answer or intermediate step to the given context. ARGUMENT & CRITIQUE TASKS MP.3 Construct viable arguments and critique the reasoning of others • Basing explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student’s response) • Distinguishing correct explanation/reasoning from that which is flawed , and – if there is a flaw in the argument – explaining what it is. For example, flawed ‘student’ reasoning is presented and the task is to correct and improve it. Tasks presenting students with common errors are valuable because students are often better at explaining why something is wrong that why something is right. CRITIQUE EXAMPLE Joan wrote “2/3 + 2/3 = 4/6” Critique her work. Is it correct? Explain why or why not. Tell what Joan did right or wrong. TOOL USAGE TASKS MP.5 Use appropriate tools strategically Diagrams as a tool: Problems that are fairly easy to solve or to answer correctly if you first draw a diagram, very hard to solve or to answer correctly if you don’t – yet no direction is given to the student to draw a diagram. (Note: MP.5 is not a coded statement to ‘use a calculator.’ If the student is not being strategic, the student is not meeting the standard. Tasks must create circumstances for poor use of tools, misuse of tools, and/or mistakenly not using tools. Sara built the block tower with 1-foot cubes. How many cubes did she use? PRECISION-ORIENTED TASKS MP.6 Attend to precision Tasks requiring the student to present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such as equals signs appropriately. For example, rubrics award less than full credit for the presence of nonsense statements such as 1 + 4 = 5 + 7 = 12, even if the final answer is correct. STRUCTURE-ORIENTED TASKS MP.7 Look for and make use of structure • Mathematical and real-world problems that reward seeing structure in an expression and using the structure to rewrite it for a purpose • Numerical problems that reward or require deferring calculations steps until one sees the overall structure. Examples such as 357 + 17999 + 1 or 37 × 25 × 4 REPEATED REASONING TASKS MP.8 Look for and express regularity in repeated reasoning • Problems in which a tedious and repetitive calculation can be made shorter by observing regularity in the repeated steps CONCEPTUAL UNDERSTANDING • • Assess where the standards explicitly call for it These should mostly be short tasks that are: computationally non-intensive and easy to answer quickly, if the student understands the concept in question, but difficult to answer at all if the student doesn’t understand the concept Examples: Write four fractions that are all equal to 5: Which number is least and which is greatest? ¾ 2 4/4 3/5 Write a number that is greater than 1/5 and less than ¼: Plot each of the following on the number line: 2 5/4 3× ½ ¾+¾ 2 – (1/10) FLUENCY Fluency assessment (with machine scoring of responses entered by computer interface) Fluency when the standards explicitly call for it Fluency priorities: 3.OA.7, 4.NBT.4, 5.NBT.5, 6.NS.2 Mark each equation as true or false 8× 9 = 80 – 8 54 ÷ 9 = 24 ÷ 6 7 × 5 = 25 8× 3 = 4 × 6 49 ÷ 7 = 56 ÷ 8 MATHEMATICAL REASONING TASKS Call for written arguments/justifications, critique of reasoning, or precision in mathematical statements (MP.3, MP.6) May include other mathematics standards as well EXPRESSING MATHEMATICAL REASONING 3.OA.5 Amber doesn’t know what 7 × 5 equals, but she knows 5 × 5 = 25 and 2 × 5 = 10. Use drawings, words, and/or equations to explain why Amber can add 25 and 10 to find what 7 × 5 equals. MODELING & APPLICATION Tasks that involve real-world contexts or scenarios and require the student to: Apply knowledge and skills articulated in specified standards; Engage particularly in the modeling practice (MP.4) Make sense of problems and persevere to solve them (MP.1) Reason abstractly and quantitatively (MP.2) Use appropriate tools strategically (MP.5) Look for and make use of structure (MP.7) and/or Look for and express regularity in repeated reasoning (MP.8) Each task consist of several related questions, or a single prompt with a longer response required, or a combination. MODELING TASKS MP.4 Model with mathematics • Contextual word problems involving ideas that are currently at the forefront of the student’s developing mathematical knowledge (e.g., multiplication or division in Grade 3 or addition and subtraction of fractions in Grade 4) • Multi-step contextual word problems in which the problem is NOT broken into steps or sub-parts • Problems of either these two kinds may involve related practices, MP.1, MP.2, MP.5, MP.7, MP.8 Grade 3 example: There are 60 straws in a package. How many straws are in 7 packages? MODELING 4th grade example of a multi-step contextual word problem in which the problem isn’t broken into steps or sub-parts A plate of cookies: There were 28 cookies on a plate. Five children each ate one cookie. Two children each ate 3 cookies. One child ate 5 cookies. The rest of the children each ate two cookies. Then the plate was empty. How many children ate two cookies? Show your work. INTEGRATIVE TASKS Integrative tasks are: • Tasks that are aligned to a cluster heading, domain heading or grade/course title, rather than to one specific standard • These tasks are not intended to permit loose interpretations of the standards. INTEGRATIVE TASKS The following problem codes straightforwardly to the cluster “Understanding place value” at Grade 5, but may not clearly code to any single standard in the cluster. Write a number in each space to make true equations. 1 tenth = 100 tenths = 0.1 tenths = 0.01 tenths = 1/10 tenths = hundredths hundredths hundredths hundredths hundredths tenths = 0.1 hundredths INTEGRATIVE TASKS 4.NF – Since this problem involves concepts and skills in fraction addition, multiplication and equivalence, it does not clearly code to any single standard in the NF domain. 9 large trucks are carrying ½ ton of lumber each. 7 small trucks are carrying ¼ ton of lumber each. How many total tons are being carried by all of the trucks? INTEGRATIVE TASKS The following grade 4 task involves place value understanding as well as elements of place value computation, so it blends the first two clusters in 4.NBT 893,462 840,924 824,595 824,162 810,930 808,879 799,982 778,877 777,852 766,398 Find two numbers in this table that differ by approximately two thousand INTEGRATIVE TASKS Each of the following three illustrations is plausible as a direct implication of the grade 5 standards for Number and Operations in Base Ten, but each involves place value understanding as well as elements of place value computation, so each blends the first two clusters in 5.NBT Mark each statement true or false. 108 × 30 is equal to a four-digit number. 6731 × 23 is equal to a four-digit number. 2244 ÷ 11 is equal to a three-digit number. INTEGRATIVE TASKS A bakery made 3,200 cookies and needs to package them in bags for a sale. 10 cookies will go in each bag. How many bags will be needed? **** Compute each of the following 357 + 17,999 + 1 357 + 17,999 899 + 1343 + 101 37 × 25 × 4 1001 × 20 RESOURCES Illustrative Mathematics http://illustrativemathematics.org Common Core Tools http://commoncoretools.wordpress.com , a blog moderated by Dr. William McCallum, distinguished professor and head of mathematics at the University of Arizona and mathematics lead for the Common Core State Standards for Mathematics. STEPS IN DESIGN 1. Choose a grade level 2. List the priority standards (using the Content Emphases by Cluster Chart and Model Content Framework). 3. Choose a priority standard or cluster, and an aligned practice standard. 4. Follow the design recommendations from the powerpoint for the practice standard(s) in question. 5. Use Illustrative Mathematics and your own curriculum materials as resources. 6. Include an answer key &/or rubric. TASK DESIGN EXAMPLE Focus for Grade 3: Numbers and Operations-Fractions Standard: Develop understanding of fractions as numbers Cluster 3.NF 3d Problem: A Letter to My Friend Some people think that ¼ is larger than 1/3. Some people think that 1/3 is larger than ¼. Which do you think is larger? Write a letter to a friend telling why you are right. Use pictures, letters and numbers , or all of these to explain your thinking. TASK DESIGN EXAMPLE Focus for Grade 5: Numbers and Operations-Fractions Standard: Use equivalent fractions as a strategy to add and subtract fractions. Cluster 5.NF 1 and 2 Problem: Mr. Jones bought a can of paint. He used 3/8 of it to paint the garage door and ¼ of it to paint a wall. A. How much paint did he use altogether? B. How much paint was left?