NRICH: Lower Primary  Be a Mathematician
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 measurement
 patterns
 sequences
 shapes
 logic
 problem solving
 strategy
 thinking critically
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Rather than having elementaryaged kids focus on specific math skills (addition, geometry, and so on), these problems are organized by criticalthinking and problemsolving themes, such as working systematically, saying what you see, and practice makes perfect. Kids are encouraged to solve a problem and then convince someone else that their solution is correct. Some problems work better as group projects, others as individual exploration, but kids can always describe their approach and share the solutions with their peers. It may take time to look through all the suggestions and teachersupport material, but it's worth it: take full advantage of it all! Navigation can be tricky; be sure to browse through the problems and then present them to kids offline. If kids want to use the online interactive features themselves, give them specific instructions and guidance. Teachers could easily link the tasks in these problems to other subjects, such as the scientific process, writing (kids describe the problemsolving process), and more.
Read More Read LessKey Standards Supported
Arithmetic With Polynomials And Rational Expressions  
HSA.APR: Perform Arithmetic Operations On Polynomials  
HSA.APR.1  Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 
Rewrite Rational Expressions  
HSA.APR.6  Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 
HSA.APR.7  (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. 
Understand The Relationship Between Zeros And Factors Of Polynomials  
HSA.APR.2  Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). 
HSA.APR.3  Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 
Use Polynomial Identities To Solve Problems  
HSA.APR.4  Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples. 
HSA.APR.5  (+) Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.1 
Counting And Cardinality  
K.CC: Compare Numbers.  
K.CC.6  Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.1 
K.CC.7  Compare two numbers between 1 and 10 presented as written numerals. 
Count To Tell The Number Of Objects.  
K.CC.4  Understand the relationship between numbers and quantities; connect counting to cardinality. 
K.CC.4.a  When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. 
K.CC.4.b  Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. 
K.CC.4.c  Understand that each successive number name refers to a quantity that is one larger. 
K.CC.5  Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects. 
Know Number Names And The Count Sequence.  
K.CC.1  Count to 100 by ones and by tens. 
K.CC.2  Count forward beginning from a given number within the known sequence (instead of having to begin at 1). 
K.CC.3  Write numbers from 0 to 20. Represent a number of objects with a written numeral 020 (with 0 representing a count of no objects). 
Geometry  
1.G: Reason With Shapes And Their Attributes.  
1.G.1  Distinguish between defining attributes (e.g., triangles are closed and threesided) versus nondefining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes. 
1.G.2  Compose twodimensional shapes (rectangles, squares, trapezoids, triangles, halfcircles, and quartercircles) or threedimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.4 
1.G.3  Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. 
2.G: Reason With Shapes And Their Attributes.  
2.G.1  Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 
2.G.2  Partition a rectangle into rows and columns of samesize squares and count to find the total number of them. 
2.G.3  Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. 
3.G: Reason With Shapes And Their Attributes.  
3.G.1  Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. 
3.G.2  Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. 
Measurement And Data  
1.MD: Measure Lengths Indirectly And By Iterating Length Units.  
1.MD.1  Order three objects by length; compare the lengths of two objects indirectly by using a third object. 
1.MD.2  Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of samesize length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. 
Represent And Interpret Data.  
1.MD.4  Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 
Tell And Write Time.  
1.MD.3  Tell and write time in hours and halfhours using analog and digital clocks. 
2.MD: Measure And Estimate Lengths In Standard Units.  
2.MD.1  Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 
2.MD.2  Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. 
2.MD.3  Estimate lengths using units of inches, feet, centimeters, and meters. 
2.MD.4  Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. 
Relate Addition And Subtraction To Length.  
2.MD.5  Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 
2.MD.6  Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent wholenumber sums and differences within 100 on a number line diagram. 
Represent And Interpret Data.  
2.MD.10  Draw a picture graph and a bar graph (with singleunit scale) to represent a data set with up to four categories. Solve simple put together, takeapart, and compare problems4 using information presented in a bar graph. 
2.MD.9  Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in wholenumber units. 
Work With Time And Money.  
2.MD.7  Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. 
2.MD.8  Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? 
3.MD: Geometric Measurement: Recognize Perimeter As An Attribute Of Plane Figures And Distinguish Between Linear And Area Measures.  
3.MD.8  Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. 
Geometric Measurement: Understand Concepts Of Area And Relate Area To Multiplication And To Addition.  
3.MD.5  Recognize area as an attribute of plane figures and understand concepts of area measurement. 
3.MD.5.a  A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. 
3.MD.5.b  A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. 
3.MD.6  Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). 
3.MD.7  Relate area to the operations of multiplication and addition. 
3.MD.7.a  Find the area of a rectangle with wholenumber side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. 
3.MD.7.b  Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning. 
3.MD.7.c  Use tiling to show in a concrete case that the area of a rectangle with wholenumber side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. 
3.MD.7.d  Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the nonoverlapping parts, applying this technique to solve real world problems. 
Represent And Interpret Data.  
3.MD.3  Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. 
3.MD.4  Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters. 
Solve Problems Involving Measurement And Estimation Of Intervals Of Time, Liquid Volumes, And Masses Of Objects.  
3.MD.1  Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 
3.MD.2  Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve onestep word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.7 
Number And Operations In Base Ten  
1.NBT: Extend The Counting Sequence.  
1.NBT.1  Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. 
Understand Place Value.  
1.NBT.2  Understand that the two digits of a twodigit number represent amounts of tens and ones. Understand the following as special cases: 
1.NBT.2.a  10 can be thought of as a bundle of ten ones — called a “ten.” b. 
1.NBT.2.b  The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones. 
1.NBT.2.c  The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones). 
1.NBT.3  Compare two twodigit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. 
Use Place Value Understanding And Properties Of Operations To Add And Subtract.  
1.NBT.4  Add within 100, including adding a twodigit number and a onedigit number, and adding a twodigit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding twodigit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 
1.NBT.5  Given a twodigit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 
1.NBT.6  Subtract multiples of 10 in the range 1090 from multiples of 10 in the range 1090 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 
2.NBT: Understand Place Value.  
2.NBT.1  Understand that the three digits of a threedigit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 
2.NBT.1.a  100 can be thought of as a bundle of ten tens — called a “hundred.” 
2.NBT.1.b  The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). 
2.NBT.2  Count within 1000; skipcount by 5s, 10s, and 100s. 
2.NBT.3  Read and write numbers to 1000 using baseten numerals, number names, and expanded form. 
2.NBT.4  Compare two threedigit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. 
Use Place Value Understanding And Properties Of Operations To Add And Subtract.  
2.NBT.5  Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 
2.NBT.6  Add up to four twodigit numbers using strategies based on place value and properties of operations. 
2.NBT.7  Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 
2.NBT.8  Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. 
2.NBT.9  Explain why addition and subtraction strategies work, using place value and the properties of operations.3 
3.NBT: Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.4  
3.NBT.1  Use place value understanding to round whole numbers to the nearest 10 or 100. 
3.NBT.2  Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 
3.NBT.3  Multiply onedigit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. 
Operations And Algebraic Thinking  
1.OA: Add And Subtract Within 20.  
1.OA.5  Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 
1.OA.6  Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). 
Represent And Solve Problems Involving Addition And Subtraction.  
1.OA.1  Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.2 
1.OA.2  Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 
Understand And Apply Properties Of Operations And The Relationship Between Addition And Subtraction.  
1.OA.3  Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 
1.OA.4  Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. 
Work With Addition And Subtraction Equations.  
1.OA.7  Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 
1.OA.8  Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �. 
2.OA: Add And Subtract Within 20.  
2.OA.2  Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two onedigit numbers. 
Represent And Solve Problems Involving Addition And Subtraction.  
2.OA.1  Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
Work With Equal Groups Of Objects To Gain Foundations For Multiplication.  
2.OA.3  Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 
2.OA.4  Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 
3.OA: Multiply And Divide Within 100.  
3.OA.7  Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. 
Represent And Solve Problems Involving Multiplication And Division.  
3.OA.1  Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 
3.OA.2  Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 
3.OA.3  Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
3.OA.4  Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?. 
Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic.  
3.OA.8  Solve twostep word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3 
3.OA.9  Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. 
Understand Properties Of Multiplication And The Relationship Between Multiplication And Division.  
3.OA.5  Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 
3.OA.6  Understand division as an unknownfactor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. 
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