We used this task for sixth graders. The objective was for students to use proportional thinking to reason out which recipe was more concentrated. Students selected one of two solutions in order to state a claim, provide evidence and state a warrant. Many students would find a common denominator in order to demonstrate a higher value ratio of corn syrup to water or total amount (sweeter) or lower value (less sweet). Students might also show how one mixture is sweeter with visual representations such as circles or a drawing of a pitcher with labeled portions.
Microsoft Word version: 6_RP_ProportionalReasoning_Problem_Critique_WhichIsSweeter
PDF version: 6_RP_ProportionalReasoning_Problem_Critique_WhichIsSweeter
I used this task with a group of 5 students that I selected for my evaluation. When I started with this group in mid/late January they were chosen by the teacher and myself to receive intervention as well. This year the students were learning about ratios and proportions. I tried to build their understanding of ratios by doing Number Talks before giving them this particular task. The students were required to use a rubric I found on the Exemplars website to help guide them in solving and writing an argument for this problem. The students were instructed to use more than one model to solve this task
Microsoft Word version: 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants
PDF version: 6_RP_ProportionalReasoning_Problem_Critique_CarlylesPlants
This task was designed for 6th graders working on condensing their writing to make their argument completely and quickly. The content is scaling figures, using proportional reasoning to prove similarity. Students who struggled with mulitiplying decimals struggled to find a multiplier. Students also struggled with providing sufficient reasoning in the brief space.
Microsoft Word version: 6_RP_ProportionalReasoning_BriefArgument_Construct
PDF version: 6_RP_ProportionalReasoning_BriefArgument_Construct
Water Sports Survey is a task developed for sixth grade students working on percents and interpretations of statistics. Students are given a table that displays two sets of data. Students must critique an interpretation made by a student. The interpretation focuses on a common misunderstanding made when looking at statistics: students often neglect to think about the whole, and instead automatically think that a larger percent means more people. Students must agree or disagree with the interpretation and justify the answer.
Microsoft Word version: 6_RP_Percents_Problem_Critique_WaterSportsSurvey
PDF version: 6_RP_Percents_Problem_Critique_WaterSportsSurvey
This problem was used by students after working visually and conceptually to explain division with fractions. They use this problem as a transition to dividing fractions using the mulitplicative inverse. Students benefitted from constructing a diagram that showed the fence sections. Constructing the diagram on graph paper, using 20 squares for the length helped students divide equally and find the fraction they needed. Explaining the connections between the diagram and the equation are very important for constructing the argument.
Microsoft Word version: 6_NS_FractionsDivision_Problem_Critique
PDF version: 6_NS_FractionsDivision_Problem_Critique
We used this task for sixth graders later in the year after working with argumentation for several months. As students became proficient with providing evidence and warrants for their own arguments, we transitioned to explaining others’ reasoning. This task required students to provide warrants to explain the evidence provided. Students needed to understand the distributive property, commutative property, and how to simplify equations. Some struggled with explaining why each step was “warranted.”
Microsoft Word version: 6_EE_ExpressionsSimplifying_ArgumentFrame_Critique
PDF version: 6_EE_ExpressionsSimplifying_ArgumentFrame_Critique
This problem is meant for students developing their understanding of the distributive property in algebraic representations. A number trick is a series of calculations in which the person giving the directions always knows what number the players will end up with. When the students are asked to represent Step 3 in a different way, they will use the distributive property. When students use parentheses, the outside number is the number of groups. The expression inside the parentheses is what is in one group. When they write the expression without parentheses, they are saying how many total x tiles and total 1 tiles they have. Switching between these two representations gives a concrete, visual, and abstract expression of using the distributive property to group and ungroup terms in an expression.
Microsoft Word version: 6_EE_ExpressionsOrderOfOperations_Problem_Construct
PDF version: 6_EE_ExpressionsOrderOfOperations_Problem_Construct
This task is designed for sixth grade students developing skills with geometric formulas for area while also working on algebraic interpretations of expressions and equations. Students are given the formula for the area of a triangle, written in two different ways. Students must use logic and algebraic knowledge to determine if the two expressions are equivalent, and why. This task highlights the relationship between multiplying by a fraction, and dividing. Students are critiquing a student interpretation.
Microsoft Word version: 6_EE_ExpressionsEquivalence_Problem_Critique
PDF version: 6_EE_ExpressionsEquivalence_Problem_Critique
This task is for sixth-grade students learning algebraic expressions. Students are asked a series of questions on how many tiles are needed to frame a picture, given a set side length. Misconceptions on these algebraic expressions are addressed in part c, when students are asked to critique the mistake made by a customer. Combining in the second extension problem brings in argumentation, as students are asked to explain how they arrived at their answer.
Microsoft Word version: 6_EE_Expressions_Problem_Critique
PDF version: 6_EE_Expressions_Problem_Critique
This task is designed for sixth graders developing skills with rate of change. The task provides students with a distance travelled along with the amount of time taken. Students must use this information to determine how far someone could get in a given amount of time. Students must recognize that a rate of change is necessary. Students may solve the problem using a rate and creating an expression, or by creating equivalent fractions, which may open the class to discussion about different methods. Students are asked to provide an explanation and evidence for why a rate is necessary, which provides an opportunity for students to create a solid argument.
Microsoft Word version: 6_EE_AlgebraRateOfChange_Problem_Construct
PDF version: 6_EE_AlgebraRateOfChange_Problem_Construct