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Final Exam Study Guide
Math Alliance Project
The exam will be given on Tuesday, August 3, 2010; you will be given two hours.
Note: The exam is worth 30% of your Math Grade and 5% of your Supporting All Learners grade.
No re-take or make-up exams will be given after this date.
Drop-in study sessions for mathematics support:
Wednesday July 28, 9:00am-11:30am
UWM - Enderis Hall 276
Thursday July 29, 9:00am-11:30am
UWM - Enderis Hall 276
Monday August 2, 5:00pm-7:00pm
UWM - Enderis Hall 276
Drop-in study sessions for Supporting all Learners topics:
Thursday July 29, 12:00-1:30
UWM - Enderis Hall 698
Monday August 2, 3:00-5:00
UWM - Enderis Hall 698
Mathematics Topics:
(1) Two natural ways to interpret division contexts:
 Partitive - known: “number of groups” unknown: “size of group”
 Measurement – known: “size of group” unknown: “number of groups”
 Know how to represent each way using words, pictures, and numbers, including posing word problems
and describing real-world contexts for each division interpretation.
Try This: Pose two word problems for 37 ÷ 4, one using a partitive context and one using a measurement
context. Identify the division context for each and explain why it is this type.
(2) Thinking multiplication to solve division problems
 Near facts with missing factors and leftovers connected to division.
Find the largest factor without going over the target number. Explain your reasoning using the language of
“groups of,” “left over,” “too high,” and “too small.”
Example: 12 × ? = 89
 12 × 10 = 120 too big
 12 × 5 = 60 too small
 Add on 2 more groups of 12 (2 × 12 = 24) to 60, that gets me to 84, 60 + 24 = 84
 So then my answer would be: 12 × 7 = 84 with 5 leftover
 89 ÷ 12 = 7 with 5 leftover
Try These: Explain how a student early on in their learning of division might use “missing factor”
reasoning to solve 6 x ? = 40, 7 x ? = 68, 4 x ? = 29

Larger numbers with missing factor reasoning. Example: 317 ÷ 7 = ?
 Restate as a missing factor: ? × 7 = 317 Think: How many groups of 7 are in 317?
 Explain your reasoning, similar to above for smaller numbers, using the language of “groups of,”
“left over,” “too high,” and “too small.”
Connect your reasoning to conceptual understanding and algorithms/strategies.
o “subtracting out” multiple same-sized groups
o partial quotient/ladder method algorithm
o partitioning the dividend.
Try These: Explain how one might use “missing factor” reasoning to solve 450 ÷ 21=? and 1041 ÷ 34 = ?.
Then show how to use the ladder method for each problem.
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(3) Concept-based definition of division
General: Encompass many situations and interpretations (not limiting to just one view)
Visualize actions on quantities (not numbers).
Accurate in the long term (don’t set students up for misconceptions).
Language attends to the conceptual meaning of the operations.
(4) Compare fractions using conceptual thought patterns and benchmarks and explain using concept-based
language to prove without a doubt which fraction is larger.
 More of the same-size parts.
 Same number of parts but different sizes.
 More or less than one-half or one whole.
 Distance from one-half or one whole (residual piece).
Try These: Which is larger: 17/43 or 17/50; 19/20 or 23/24; 22/7 or 15/8; 13/14 or 25/28.
(5) Demonstrate “operation sense” related to adding and subtracting fractions through contextual situations
 Justify your thinking when adding and subtracting fractions using concrete models and estimation
strategies.
 Demonstrate your ability to make reasonable estimates for fraction computation based on
benchmarks and conceptual thought patterns.
Try These: Is the answer to 3/8 + 2/17 more or less than one-half? Pose word problems, solve using
diagrams, explain, 3/8 + 3/4 = ?, 2 7/9 + 2/3 = ?,
1 1/8 – ½ = ?, 2 ¼ – 7/8 = ?
(6) Demonstrate “operation sense” related to multiplication and division of common fractions through contextual
situations.
 Pose word problems to contextualize multiplication and division with fractions and draw diagrams that
illustrate solution paths. Also identify the role of each number in the problem such as representing the
“number of groups,” the “size of a group,” and the total amount, and the unit whole.
 Interpret the “remainder” in division situations as related to the “number of groups” or the “size of a
group,” and how this relates to “leftover” type reasoning.
Try These:
• Pose word problems, solve using diagrams, explain, 4 x 3/8, 2 ¼ x 4/5, 4/9 x 3
• Pose word problems and solve using diagrams, interpret any remainders as related to groups and as
related to leftovers: 5 1/3 ÷ 2/3 = ?, 3 ÷ 3/5 = ?, 9 ÷ 4 = ?, 9/10 ÷ 3 = ?,
7/8 ÷ 3/4 = ?, 1 2/3 ÷ 4/9 = ?
Supporting All Learners Topics:
You will select a math activity we have done in class from the following list:
 Comparing fractions using fraction cards and benchmark cards
 Fraction strips (the creation of the strips or the use of the strips for adding, subtracting, or comparing fractions)
 Estimating fraction operations with benchmarks
 Lima bean activity
We will provide descriptions of students with specific difficulties in language (receptive/expressive or both), visual
spatial processing, or behavior. You will then select two of the students, identify their ages (your choice), and:
 Identify specific challenges they may have with the math activity
 Describe how you would structure the math activity when teaching the students in a group setting
 Describe the kinds of scaffolds you would provide to address the students’ particular difficulties.
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